KARA   ARITHMETICA 


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Plate  I.    From  a  manuscript  of  boethius,  c.  1294,  showing 

FIOUKATE    NUMBERS 


RARA    ARITHMETICA 

OLA  CATALOGVE  OF  THE  ARITHMKTICS 
WRITTEN  BEFORE  THE  YEAR  MDCT  WEFH  A 
DESCRIPTION  OF  THOSE  IN  THE  LIBRARY  OF 
GEORGE  ARTHVR  PLIMPTON  OF  NEW  YORK 
BY  DAVID  EVGENE  SMITH  OF  TEACHERS 
COLLEGE  COLVMBIA  VNIVERSITY 


GINN  AND   COMPANY  PVBLISHERS 
BOSTON  AND  LONDON   MDCCCCVIII 


COPYRIGHT,  1908,  BY  DAVID  EUGENE  SMITH 
ENTERED  AT  STATIONERS'  HALL 


GE)VEH^ 


GINN   &   COMPANY  •   PRO- 
PRIETORS •  BOSTON  •  U.S.A. 


QAio 


I  ■ 
Ai^S'ss 


VIRO 

DOCTISSIMO  •  ILLVSTRISSIMO 

GEORGIO  •  ARTVRO  •  PLIMPTON 

ARTIVM  •  MAGISTRO 

LIBRORVM  •  VETVSTIORVM  ■  AMATORI 

AVCTORVM  ■  AMICO  •  SOCIO  •  ERVDITORVM 

LITTERARVM 

FAVTORI  •  FOTORI 


195104 


PREFACE 

One  of  the  first  and  most  important  questions  for  the  student 
of  mathematical  history  is  that  relatin^^  to  the  available  sources  of 
information.  In  the  fields  of  higher  mathematics  scholars  have 
been  more  or  less  successful  in  bringing  together  these  sources, 
and  in  listing  them  in  bibliographies  ;  but  in  that  humbler  field  in 
which  primitive  mathematics  first  found  root,  only  a  few  biblio- 
philes have  sought  to  preserve  the  original  material,  and  no  one 
has  seriously  attempted  to  catalogue  it.  Libri,  it  is  true,  brought 
together  two  large  libraries  of  Rara  Arithmetica,  but  he  was 
neither  a  true  book-lover  nor  a  true  scholar,  for  he  gathered  his 
treasures  purposely  to  see  them  dispersed,  his  commercial  spirit 
scattering  at  random  what  should  have  been  kept  intact  for  the 
use  of  scholars.  Prince  Boncompagni,  the  most  learned  of  all 
collectors  in  this  domain,  lived  to  see  an  unappreciative  city 
ignore  his  offer  to  make  his  magnificent  library  permanent,  and 
at  his  death  it  was  scattered  abroad,  as  had  been  the  lesser  ones 
of  Kloss  and  De  Morgan.  The  third  great  collection  of  early  text- 
books which  has  been  made  in  recent  years  is  that  of  Mr.  Plimp- 
ton. Of  the  libraries  of  arithmetics  printed  before  the  opening  of 
the  seventeenth  century  his  is  the  largest  that  has  ever  been 
brought  together,  not  excepting  Boncompagni's,  and  it  may  well 
be  doubted  if  another  so  large  will  again  be  collected  by  one  man. 
De  Morgan  was  able  to  examine,  in  the  British  Museum  and  else- 
where, less  than  a  hundred  arithmetics  written  before  1601, 
including  all  editions  ;  but  Mr.  Plimpton  has  more  than  three 
hundred,  a  number  somewhat  in  excess  of  that  reached  by  Bon- 
compagni. Indeed  there  are  few  arithmetics  of  much  importance 
that  are  not  found,  in  one  edition  or  another,  in  his  librarv. 


X  KARA   ARITHMETICA 

The  writers  of  these  early  printed  books,  not  all  themselves 
of  the  centuries  under  consideration,  were  by  no  means  obscure 
men.  Among  them  was  Boethius,  whom  Gibbon  called  "  the  last 
of  the  Romans  whom  Cato  or  Tully  could  have  acknowledged 
for  their  countryman."  In  the  list  are  the  names  of  Cassiodorus 
and  Capella,  who  at  least  represented  what  there  was  of  culture 
in  their  day,  and  Isidorus,  the  learned  Bishop  of  Seville.  There 
are  also  the  names  of  Archimedes,  who  deemed  it  a  worthy  labor 
to  improve  the  number  system  of  the  Greeks  ;  of  Euclid,  whose 
contributions  were  by  no  means  confined  to  geometry  ;  of  Nicoma- 
chus  and  lamblichus,  who  represent  the  declining  Hellenic  civiliza- 
tion, and  of  Psellus,  who  was  a  witness  of  its  final  decay.  There, 
too,  are  the  names  of  the  Venerable  Bede,  of  Sacrobosco,  and  of 
Bradwardin,  all  of  whom  testify  to  the  culture  of  mediaeval  Eng- 
land ;  of  that  great  Renaissance  compiler,  Paciuolo ;  of  Tartaglia 
and  Cardan,  who  helped  to  make  the  modern  algebra,  and  of  such 
scholars  as  Ramus,  Melanchthon,  and  Bishop  Tonstall.  Worthy 
as  such  a  list  may  be,  it  is  rendered  none  the  less  so  by  the  names 
of  Widman,  Kobel,  Borghi,  Riese,  and  Gemma  Frisius,  mere  arith- 
meticians though  they  were,  for  few  who  read  their  works  can  fail 
to  recognize  that  they  powerfully  influenced  education,  not  only 
in  their  own  time  but  for  generations  after  they  had  passed  away. 

In  view  of  the  fact  that  the  fifteenth  and  sixteenth  centuries 
constituted  the  formative  period  in  the  history  of  printed  arith- 
metics, I  have  felt  it  a  not  unpleasant  duty  to  catalogue  the  volumes 
in  Mr.  Plimpton's  library  that  belong  to  this  period  and  subject, 
including  such  later  editions  as  it  may  contain,  —  to  give  a  brief 
statement  of  their  contents,  and  to  supplement  this  work  by  a  list 
of  other  arithmetics  published  before  1 60 1 .  That  a  complete  bibli- 
ography is  impossible  is  evident  to  any  one  who  considers  the 
subject.  It  is  a  simple  matter  to  consult  the  few  lists  of  early 
mathematical  works,  and  to  trace  the  names  thus  secured  through 
such  catalogues  as  those  of  the  British  Museum  and  the  Biblio- 
theque  nationale  (unfortunately  only  just  begun),  and  through 
bibliographies  like  those  of  Gra'esse  and  Hain  and  Coppinger.    It 


PREFACE  xi 

is  also  an  easy  matter  to  examine  the  masterly  work  of  Riccardi, 
the  less  accurate  list  of  Murhard,  the  catalogues  of  Libri,  and 
numerous  other  works  of  a  similar  nature.  But  it  is  manifestly 
impossible  to  read  all  of  the  published  catalogues,  almost  invari- 
ably arranged  only  by  authors.  Therefore  many  extant  books 
will  necessarily  remain  undiscovered,  and  it  is  probable  that  some 
will  always  elude  the  eye  of  the  special  bibliographer.  Such  is 
the  work  that  has  been  done  in  preparing  this  volume,  and  such 
is  the  feeling  of  insufficiency  of  achievement  that  remains.  There 
is,  however,  a  satisfaction  in  knowing  that  the  bibliography  is 
based  in  large  measure  upon  an  examination  of  the  books  them- 
selves in  various  libraries,  and  that  the  secondary  sources  are  of 
recognized  authority.  Over  five  hundred  and  fifty  different 
works  are  mentioned,  or,  including  the  various  editions,  nearly 
twelve  hundred  books  in  all.  Of  the  five  hundred  and  fifty  books, 
about  four  hundred  and  fifty  are,  strictly  speaking,  arithmetics. 
Of  all  arithmetics  known  to  have  been  printed  in  the  sixteenth 
century,  and  to  have  been  important  enough  to  have  two  or  more 
editions,  Mr.  Plimpton's  library  lacks  less  than  twenty-five.  Of 
those  which  were  published  but  once,  some  are  known  only  by 
name,  while  the  rest  are  mostly  '  abachetti  '  or  '  Rechenbuch- 
leins  '  — mere  primers  of  a  few  pages  and  of  no  importance. 

It  must  also  be  borne  in  mind  that  it  was  inevitable  that  cer- 
tain arithmetics  of  the  sixteenth  century  should  have  perished 
utterly,  leaving  not  even  a  record  of  their  existence.  Their 
very  commonness  often  caused  their  destruction,  a  law  of  which 
the  unique  surviving  copy  of  more  than  seventy  thousand  New 
England  Primers  of  the  Franklin-Hall  press  is  a  lonely  witness 
in  our  country. 

It  is  inevitable  that  there  should  be  errors  in  such  a  list. 
Titles  have  been  included  when  mentioned  in  even  one  standard 
bibliography,  although  they  cannot  be  found  in  others,  it  being 
practically  impossible  for  one  person  to  verify  every  item.  It  is 
hoped,  however,  that  a  foundation  has  been  laid  upon  which 
others  may  build,  eliminating  or  otherwise  as  the  case  may  be. 


xii  RARA   ARITHMETICA 

Of  other  works  in  this  field,  Httle  need  be  said.  De  Morgan's 
Arithmetical  Books  is  still  one  of  our  best  single  sources, 
although  sixty  years  have  elapsed  since  it  first  appeared.  De 
Morgan,  however,  mentions  altogether  about  one  hundred  and 
twenty  editions  of  works  originally  appearing  before  1601,  against 
nearly  twelve  hundred  listed  here.  Unger  and  Tropffke  are 
both  scholarly  writers,  but  their  bibliographies  are  almost  exclu- 
sively German.  Sterner  was  less  of  a  student,  and  his  list  is  cor- 
respondently  less  valuable.  Historians  like  Cantor  and  Zeuthen 
have  given  this  particular  period  only  nominal  attention,  save  as 
to  a  few  great  arithmeticians,  ignoring  those  contributions  which 
set  forth  the  real  work  of  the  people's  schools.  The  titles  of  the 
works  of  such  writers  have  not  been  given,  since  any  reader  of  a 
bibliography  like  this  will  know  the  more  general  histories. 

Mr.  Plimpton's  library  has  also  a  number  of  valuable  manu- 
scripts on  arithmetic.  Since  these  are  not  available  for  students 
generally,  although  of  great  value  in  themselves,  they  have  been 
placed  after  the  printed  works  instead  of  being  inserted  in  chrono- 
logical sequence.  A  study  of  our  numeral  system  has  justified  the 
inclusion  of  books  which,  if  printed,  would  hardly  have  place. 
Manuscripts  written  before  the  forms  of  numerals  were  fixed  are 
often  valuable  in  tracing  their  development,  even  though  the 
books  themselves  are  not  arithmetical.  Only  those  have  been 
catalogued  which  bear  with  some  directness  upon  arithmetic,  and 
which  were  written  before  the  year  1601,  although  numerous 
others,  in  many  respects  as  valuable,  and  including  several  inter- 
esting works  on  the  calendar,  the  sphere,  and  astrology,  are  in 
the  library. 

One  difficulty  attendant  upon  a  labor  of  this  kind  is  to  deter- 
mine what  printed  books  to  include.  The  number  might  easily 
have  been  increased  by  listing,  as  De  Morgan  occasionally  did, 
works  that  are  not  at  all  arithmetical  —  Peletier's  algebra  for 
example.  It  has  been  thought  better  to  draw  the  line  in  gen- 
eral more  closely  than  he  did,  and  to  depart  from  genuine  arith- 
metics only  in  the  case  of  the  works  that  discuss  at  least  some 


PREFACE  xiii 

questions  relating  to  the  science  or  art  of  numbers.  No  effort 
has  been  made  to  add  to  the  supplementary  lists  books  that 
are  not  purely  arithmetical,  such  as  treatises  on  the  ancient 
measures,  although  those  that  are  in  Mr.  Plimpton's  library,  and 
are  of  value  in  the  study  of  the  history  of  arithmetic,  have  been 
included. 

The  arrangement  is  chronological  by  first  editions,  but  the  Index 
allows  for  alphabetical  and  geographical  reference.  Although  the 
nature  of  the  work  is  usually  discussed  very  briefly,  an  examina- 
tion of  the  Index  will  show  that  a  fairly  complete  history  of 
Renaissance  arithmetic  has  been  included  in  the  work  —  a  his- 
tory which  I  hope  to  present  in  other  form  in  the  future.  The 
uncertainty  in  the  use  of  such  symbols  as  4°  and  8"  has  led  to 
the  measuring  of  the  page  and  text.  The  page  varies  owing  to 
the  binder's  work,  and  the  text  is  not  uniform  page  for  page,  but 
this  plan  seems  the  most  satisfactory  one  for  giving  the  size  of 
the  book.  The  centimeter  has  been  taken  as  a  unit  of  measure, 
since  all  English  and  American  readers  of  a  work  like  this  will 
be  familiar  with  it,  while  our  popular  units  would  be  unknown  to 
most  others.  These  measures,  like  the  number  of  lines  to  a  page, 
of  course  vary  considerably  in  the  same  book.  The  statement 
*  There  were  no  other  editions  '  is  to  be  understood  to  mean  that 
I  have  found  no  others  that  were  printed  before  1601.  The 
illustrations  have,  in  general,  been  selected  with  a  view  to  biblio- 
graphical needs,  although  many  have  a  marked  historical  interest. 

In  copying  the  titles  it  has  been  the  intention  to  follow  the 
original  as  closely  as  possible,  without  attempting  to  imitate  par- 
ticular forms  of  type  or  to  use  capitals  except  as  initials.  In  the 
cases  of  misspelled  words,  omitted  capitals,  and  peculiar  punctua- 
tion, the  errors  have  been  copied  as  faithfully  as  possible.  At 
the  same  time  mistakes  must  have  been  made  in  transcribing, 
although  it  is  hoped  that  they  are  not  of  a  serious  nature. 

DAVID   EUGENE  SMITH 


LIST  OF  PLATES 


PAGE 

Plate  I.    From  a   manuscript  of  hof.thhis,  c.   1294,  show- 
ing figu  rate  numbers        Frontispiece 


Plate  II.    From  the  marcakita  i-hilosoimhca 82 

A.  Geometry               B.  Arithmetic- 
Plate  III.    Title  page  of  lax 122 

Plate    IV.    From   the    campanus    manuscript    of   euclid, 

c.   1260 433 

Plate  V,    From  a  manuscript  of  euclid,  c.   1294  ....  436 

Plate  VI.    From  a  manuscript  of  boethius,  c.   1300      .     .  440 

Plate  VII.    From   a  manuscript  of   giovanni    da    firenze, 

1422 446 

Plate  VIII.    From  a  manuscript  of  sacrobosco,  c.   1442     .  450 
Plate  IX.   From  an  anonymous  manuscript,  c.   1460      .     .  462 


.      ABBREVIATIONS 

c,  circa,  about 

cm,,  centimeters 

ed.  pr.,  edito  priuceps,  first  edition 

f.,  ff.,  folio,  folios 

fol.,  4°,  8°,  .  .  .,  folio,  quarto,  octavo,  .  .  . 

ib.,  ibidem,  the  same  place 

1.,  11.,  line,  lines 

p.,  pp.,  page,  pages 

r.,  recto,  the  first  page  of  a  leaf 

s.  a.,  sine  anno,  without  date  of  publication 

s.  1.,  sine  loco,  without  place  of  publication 

s.  1.  a.,  without  place  or  date  of  publication 

v.,  verso,  the  second  page  of  a  leaf 

//,  the  end  of  a  line  of  print 


PART  I 
PRINTED    BOOKS 


PRINTED   BOOKS 


ANONYMOUS.    Ed.  pr.  1478.  Treviso,  1478. 

Title.  '  Incommincia  vna  practica  molto  bona  et  vtilc//a 
ciafchaduno  chi  vuole  vxare  lartc  dcla  niercha-/Alantia.  chia- 
mata  vulgarmcnte  larte  dc  labbacho.'    (F.  i,  r.    See  l^lg.  i.) 

Colophon.  'A  Triuifo  :  :  A  di  .io.  Deceb:^  : :  .1478.'  (V.  62,  r. 
See  Fig-.  4.) 

Description.  4°,  14.6x20.5  cm.,  the  text  being  7.3  x  12.8 
cm.    62  ff.  unnumb.,  32  11.    Treviso,  1478. 

Editions.    There  was  no  other  edition. 

So  far  as  known  this  is  the  first  practical  arithmetic  to  api)ear  in 
print,  for  Albert  of  Saxony's  Tractatus  (p.  9)  and  the  Ars  Numerandi 
(p.  23),  even  if  earlier,  and  the  Etymologies  of  Isidorus  (p.  8),  are  not, 
strictly  speaking,  of  this  class. 

The  author  of  the  book  is  unknown,  but  from  the  oi)ening  lines  it 
seems  that  he  was  a  teacher  of  arithmetic  in  Treviso.  The  i)rinter  is 
also  unknown,  although  it  was  probably  one  Manzolo,  or  Manzolino. 
The  history  of  the  work  has  been  carefully  studied  by  Boncompagni, 
his  results  appearing  in  the  Atti  dell''  Accademia  Poiiiificia  de'  Nuoin 
Lincei,  vol.  16.  This  particular  copy  was  in  the  Pinelli  collection,  and 
was  sold  on  February  6,  1790,  to  a  Mr.  Wodhull.  It  afterward  found 
its  way  into  the  library  of  Brayton  Ives,  Esq.,  of  New  York,  and  at  the 
sale  of  that  library  was  acquired  by  Mr.  Plimpton. 

The  work  is  commercial  in  character,  the  fundamental  processes 
being  taken  up  in  the  common  order,  and  these  being  followed  by  the 
rule  of  three.  A  curious  feature  not  at  all  common  in  early  arithmetics 
is  the  rule  of  two.  The  practical  applications  are  chiefly  included  under 
the  rule  of  three  and  partnership.  There  is  also  a  brief  treatment  of  the 
calendar,  for  Church  puri)Oses.  The  book  is  lacking  in  applications 
to  exchange,  and  probably  on  this  account  it  did  not  appeal  to  the 
merchant  class  sufficiently  to  warrant  a  second  edition. 


RARA   ARITHMETICA 


In  considering  these  early  works  it  is  necessary  to  understand  the 
four  types  of  arithmetics  which  the  Renaissance  inherited  from  the 
Middle  Ages.    These  types  are  as  follows  : 

I.    The    theoretical  books.    These   works   were    based    chiefly  upon 


9^ 


3fncomminda  vna  p?acti«  moltobona  et  utiles 
a  nafcbaduno  cfci  ruole  t5>;are  lai  te  tdc la  nu rcta* 
tlamia«chiamata  vulsjamif  iite  lartc  xx  labbacho*        ,[ 

■Rf^ato  ptu  e'piu  rolte  t)a  alcTrunf 
50U3ni  ami  molco  cilecaflimi : U 
quaU  pjctendeuano  a  xomx  t;oltr 
fare  la  mercbadantia:cbc  per  \oro 
amojc  me  piaceffe  affadigarmc  t> 
1  pHOc}36«e  tjargli  in  fcrtito  qualcbe  fnndaiTifto 
cerca  larte  xx  anrmetricawrbianiata  uulgarmente 
iabbacbo.Unde  w  conftretto  per  atnoi  u  I020:  rt 
caaditfad  ptilitacj  run  cbi  p:cttndi;no  a  q»ella;rc 
gondola  picola  mtelligenna  'oel  msegno mio:bO 
deliberato  fe  non  m  tuto:in  parte  tanic  fanffjre  ai 
loJO.aeio  cbe  I020  virntofi  cefideru  rale  frutto  n 
ceuere  poflfeano.^n  nome  xh  ■oio  adoncba :  tosLo 
per  pjicipio  mio  el  citto  tx  algojifmo  coft  ricedo. 
t  •  Ote  quelle  core:cbe  ra  la  p:ima  o:!i8ine 
bano  babuto  pioducnnf to:per  rarone  «e 
nuniero  fono  fta  formade..e  coli  come  fo/ 
tiotbano  -oa  fir  cognofaidr-pero  nc  la  cognmone 
cetute  lecofetqucftapjflctica  e  nrccffaria .  £^er 
intrarnel^pofitomio:prmio  fapi lccto?e-^be  qn/ 
xo  fa  9I  pjopofito  noftrotrflutntro  c  tna  molntu* 
dine  congregata  oiicro  nifcmbJada  tsa  moltc  rni^ 
tade.etal  menora  xo  vnitade^ome  e.z.  el  quale 
c  lo  pjimo  e  menoje  nunierotcbe  fe  mjona.ia  v* 
Hitffde  e  quella  cofa :  w  la  qiicle  Of  m  cofa  fi  xim. 
rna.Segodano  fapucb*  fe  truoua  numcn  xt  tre 
fnanicre.^1  pjinio  fe  cbiama  numerofimphce.lal 
tro  numcro  aroculo .  41  teno  fe  cbwma  npmcr© 


9^ 


/        i      ^ 


51- 

c 


G^ 


n 

11 

JL 

J^' 

A. 

J^^X 

-f 

r; 

Jt  A. 

^1 

Fig. 


First  page  of  the  trkviso  arithmetic 


Boethius  (p.  25),  who  wrote  in  the  beginning  of  the  sixth  century,  fol- 
lowing the  Greek  models  of  Nicomachus  (p.  186)  and  Euclid  (p.  11). 
They  are  devoted  to  such  matters  as  the  theory  of  figurate  numbers,  of 
which  the  square  and  cube  are  all  that  now  remain  in  elementary  text- 
books, and  the  cumbersome  Greek  ratio-system  which  testified  to  the 
ancient  difficulty  with  fractions.    Books  of  this  class  were  written  by 


PRINTKI)    BOOKS  5 

such    men    as    Hraawarclin    (p.    6,),    Albert   of    Saxony    (p.   9).    and 
Jordanus  (p.  62). 

2.   77^^    a(^oris,ns   (a/i^onsm,).    These   were    practical    arithn^-tics, 

aoglio  perocT^e m  mtmai che fono  aTtri nio^fj  -oe 

molnphcare  per  fcacbierotli  qiiali  kffaro  flf  Itiidi 

o  tuotmettenda  !i  epempli  fot  folaniente  m  fbima* 

come  po2ai  vedere  qui  focto 

O2  togti  w  fareropiedictofcacbiero.joe.^  14. 

fifl.9  ^  4.e  nota  w  farlo per  U  quatro  modi  comr 

qai^fotto. 


ir^4 


^'4i« 
Tp^olzis 

i  9  5  i'  ^  6 


y    9  ^ 

ri/io/ 

xl/-''/  9 
|o/!oX 

9IZ9JX5 


i  /I 
/4l< 


%        1        6 


^        ^        4 

^-9|\5|\4( 
oX|oXlo\|i 


xXioVliXN 


ecwmu   z        9 


Fig.  2.    From  the  treviso  arithmetic,  showing  various  forms 

OF    MULTIPLICATION 


RARA   ARITHMETICA 


360011 


I  I  3^96-79  5  3  3  565ooo 
5^54539^2662540 


6^X3i7Xo5t6^4$2^$i60| 
^yij-titxpjJ^jorjj^j^tjyt/f  1^10003x0 

fi^X99^gr0ifo^i  p  pop  p  pi  31 16  ^ 

fi^Xp9'^PJ^S  pop 

Jft  e  compita  dnat:  t  vi  nuto  lo  impJonnfro«50c  it 
p}r):ionun5onado(i.XbrfelirK  146  1  6  cii5e  9 
fa5i  5«g  '   valiffc  duc«t90  3  gup  M3453i% 
3         _ji^  4^io^64 

c]:^  Iirej  o  o  o*e  5  t^leranno  ducana  3  o « g .  6* 
It  quad  fcno  rn  quarto  ^e  rno  oucdto«]BicIx  qlie 
ra5one  e  qnefle  ftano  feguramente  bene* 
^uifando  te.cbe  qnando  bauerai  ^a  fare  qaalcbe 
raj:one  x>a  impojtantia  :e  cfcc  tu  wibin :  non  po2ai 
pjuouare  piu  fcgurametctcbe  polc^re  la  toa  rapo^ 
ne.al  inodo  cbe  bai  t?illo  nc  le  trr  rapoe  piedim', 
Unde  per  quelle  e  pe r  k  altre  racone  pjeditte :  fe 
i|uale  fono  in  taco  nuinero  qumd^pettii  puo  i\mn 

Fig.  3.     From  the  treviso  arithmetic,  showing  multipication 

AND   the   galley    FORM    OF    DIVISION 


PRINTI-:!)    IJOOKS  7 

written  to  supply  the  malheniatic  ;il  knowledge  necessary  for  business 
computations,  and  using  the  Hindu-Arabic  numerals.  These  numerals 
were  known  in  India,  without  the  zero,  as  early  as  the  third  century 
K.  c.  They  were  gradually  i>erfected,  and  by  the  time  they  reac  hed 
Bagdad  from  India,  in  the  eighth  century  a.  d.,  they  included  the  zero. 
An  arithmetic  employing  these  numerals  was  written  about  800  a.  d. 
by  an  Arab  scholar,  Mohammed  ibn  Musa,  known  by  the  name  of  al- 
Khowarazmi  (from  his  birthplace,  Khwarazm),  and  from  the  Latinized 
form  of  his  name  came  the  word  a/i^orism. 

3.  The  abacus  arithmetics.  1  hese  were  also  commercial  books  ;  but 
since  they  used  the  Roman  numerals,  which  were  not  suited  to  com- 
putation, the  actual  calculations  were  carried 

on  by  means  of  calculi  (Latin,  pebbles) ,  {^v/rv/.v  ou.(K..torte.(in.« 
(French  for  things  thrown  or  cast,  from  the 
Latin  jacere,  to  throw),  or  counters,  from 
which  English  form  we  have  expressions  like 
'  cast  an  account.'  The  table  on  which  the 
calculi  were  cast  is  still  called  a  counter  in  our 
shops,  but,  like  the  sand  tables  used  for  com- 
puting, it  was  in  early  times  called  an  abacus. 
Just  at  the  opening  of  the  Renaissance  the  con- 
test was  still  waging  between  the  algorists  and 
the  abacists.    These  arithmetics  are  not  found,  xb.!uou^i.«™a^Nn<«j:,.(ridK..- 

however,  in  Italy,  because  the  merchants  of        u^^'^Bi)^^Q-^7f. 
that  country  abandoned  the  use  of  the  counters   Yh-    a      I  ast   pacf    of 
long  before  this  was  done  in  other  countries.  -j.^^.  iKp^viso  arith- 

In  Germany  the  arithmetics  frequently  have  ^n•■.TIC 

in  their  titles  the  expression  *auff  der  Linien 

und  Federn '  (Riese,  p.  138,  1522),  or  '  mit  der  ziffer  unnd  niit  den 
zal  pfenningen '  (Rudolff,  p.  151,  1526),'  auff  der  Linien'  referring  to 
the  lines  on  which  the  counters  ('  zal  pfenningen ')  were  cast,  and  the 
'  Feder '  referring  to  the  pen  with  which  a  figure  (' zifTer ')  was  written. 

4.  The  computi.  The  computus  or  compotus  w-as  a  treatise  upon 
the  Church  calendar,  containing  such  simple  directions  as  were  tieces- 
sary  for  computing  the  dates  of  Easter  and  the  other  movable  feasts. 
The  chapter  on  the  calendar,  of  which  there  is  still  some  trace  in  our 
arithmetics,  originated  in  the  comjxitus.    (See  Anianus,  p.  31,  1488.) 

The  Treviso  arithmetic  is  a  good  example  of  an  algorism.  Fig.  3 
shows  that  multiplication  was  performed  as  it  is  to-day  (but  see  Fig.  2), 
and  that  division  was  ])erformed  by  the  '  galley  '  method,  so  called  because 
the  work  resembled  in  form  an  ancient  galley  with  its  sails  set. 


8  KARA   ARITHMETICA 

ISIDORUS  OF  SEVILLE.     Ed.  pr.  1472.         Venice,  1483. 

Born,  probably  at  or  near  Cartagena,  c.  560  or  570 ;  died  at  Seville,  April  4, 
636.  One  of  the  most  learned  men  of  his  time.  Bishop  of  Seville,  and  writer 
on  theology,  philosophy,  and  the  general  learning  of  the  Middle  Ages. 

Title.  'Cincipit  epiftola  Ifidori  iunioris  hifpalenfis  epi-//fcopi 
ad  Braulionem  cefar  auguftanu  epifcopu.'     (F.  i,  r.) 

With  this  is  :  '  Cin  chrifti  nomine  incipit  Hber  primus  fancti  // 
Ifidori  hifpalenfis  epifcopi  de  fumo  bono.//  Qd  deus  fumus  2 
incomutabilis  fit  CCap.  I.'    (F.  i,  r.,  following  the  first  f.  105.) 

Colophon.  '  <LFinit  liber  etymologiarum  //  Clfidori  hifpalenfis 
epifcopi.'    (F.  10 1,  v.) 

Colophon  of  the  second  part.  '  CFinit  liber  tertius  1  vltim^  de 
sijmo  bono  //  fancti  Ifidori  hyfpaleniis  ejji  :  Impreffus  //  Venetijs 
per  Petru  lofiein  de  Langenceii.//  C.M.cccc.lxxxiij.C,'  (F.  28,  r.) 

Description.  Fol.,  19.8  X  27.7  cm.,  printed  in  double  columns, 
each  being  G.y  X  23  cm.  i2gfi.  numb.  +  4  unnumb.  =  133  ff., 
58  11.    Venice,  1483. 

Editions.  Augsburg,  1472;  Venice,  1483  (here  described); 
ib.,  1485  (.'');  ib.,  1493,  fol.;  Basel,  1577  (mentioned  below).  Also 
two  editions  s.  1.  a.  (Strasburg,  1470  .?),  and  one  s.  1.  a.  (Cologne, 
1476-78  ?). 

This  book  of  etymologies  written  by  Isidorus,  Bishop  of  Seville  in  the 
seventh  century,  is  the  standard  authority  upon  the  state  of  learning  in 
Spain  in  that  period.  The  subject  of  arithmetic  is  treated  in  book  3, 
beginning  (f.  15)  :  'Cincipit  liber  tertius //CDe  vocabulo  arithmetice 
//  difcipline  CCap.  I.'  The  work  consists  entirely  of  the  mediaeval 
theory.  The  treatment  is  very  brief  (5  ff.)  and  is  followed  by  a  few 
pages  on  the  calendar.  Although  appearing  in  1472,  this  cannot  be 
called  the  first  printed  arithmetic,  since  it  touches  so  briefly  upon  the 
subject.     It  has  therefore  been  placed  after  the  Treviso  arithmetic. 

ISIDORUS   OF   SEVILLE.     Ed.  pr.  1472.  Basel,  1577. 

Title.  '  Isidori  //  Hispalensis  //  Episcopi  //  Originum  libri 
viginti/ex  antiquitate  eruti.//  .  .  .  Basileae,//per  Petrum  Pernam.' 
(F.  I,  r.) 

Description.  P'ol.,  20.5  X  31  cm.,  printed  in  double  columns, 
each  being  8.2  X  24.4  cm.,  60  11.  With  an  edition  of  Capella. 
Basel,  1577. 


PRINTI^I)    1U)()KS  9 

ALBERT  OF  SAXON  V.   Ed.  pr.  c.  1478.     Venice  (?),  c.  1478. 

]>oin  r.  1330.  He  lectured  in  the  University  of  Paris,  was  Rector  at 
\ieniui  ill  1365,  and  was  l>isii(>i)  of  Halherstadt  from  1366  to  1390.  lie 
wrote  several  scientific  works. 

Title.  '  Eccelletiffimi  magistri  albcrti  de  //  faxonia  Iractalus 
pportionum  incipit  felicitcr.'     (F.  i,  r.) 

Colophon.  '  Explitiunt  pportiones  mat^iflri //'alberti  de  faxonia.' 
(F.  9,  r.) 

Description.  Fol.,  19.7  X  28.1  cm.,  in  double  columns,  each 
being  5.6  X  16.7  cm.  9  IT.  unnumb.,  39  11.  S.  1.  a.  (Venice  .^ 
c.  1478). 

Editions.  S.  1.  a.  (Venice  .^  c.  1478,  here  described);  Padua, 
1482  (12  ff.);  ib.,  1484  (12  ff.);  ib.,  s.  a.,  but  before  1487; 
Venice,  M.cccc.xxxlvii  for  1487  (10  ff.);  ib.,  1494  (10  ff.) ;  ib. 
(with  another  work),  1496;  ib.,  1496  (but  no  copies  extant  })  ; 
Bologna  (with  another  work),  1502;  ib.,  1506  (commentary  by 
Vittori);  Paris,  s.  a.  An  epitome  by  Padre  Isidore  Isolani 
Milanese  was  published  in  Pavia  in  15 13,  and  again  in  1522. 
Prince  Boncompagni,  in  an  elaborate  discussion  of  the  various 
editions  {Bulletino,  IV,  498),  mentions  this  rare  first  edition,  of 
which  he  knew  but  one  other  copy,  that  in  the  Biblioteca 
Ambrosiana  at  Milan.  He  was  of  the  opinion  that  it  was 
printed  in  Venice,  in  a  type  used  before    1480. 

Although  the  date  is  uncertain,  this  Tractatus  may  contest  with  the 
Treviso  arithmetic  (p.  3)  and  the  Ars  Numerandi  (p.  23)  the  honor  of 
being  the  first  printed  work  devoted  wholly  to  some  phase  of  arithmetic. 
It  treats  of  ratios  according  to  the  cumbersome  method  of  Boethius  as 
followed  during  the  Middle  Ages.  It  is  purely  theoretical  and  repre- 
sents the  university  treatment  of  scientific  arithmetic  in  that  period. 

Other  7c>orl's  0/14/2-/480.  The  mathematical  activity  in  Italy  dur- 
ing this  period  was  very  considerable.  From  1472  to  1480  there  were 
38  mathematical  works  printed  in  the  country.  In  the  next  decade 
there  were  62  and  in  the  next  100,  with  13  of  uncertain  date  between 
1472  and  1500,  making  a  total  of  213  appearing  in  a  period  of  less 
than  thirty  years. 

Before  the  Treviso  book  there  were  printed  at  least  three  works  which 
touched  briefly  upon  arithmetic.     These  were  (i)  '  De  re  militari  libri 


lo  KARA   ARITHMETICA 

XII,'  by  Robertus  Valturius  of  Rimini,  printed  at  Verona,  1472,  fob,  in 
the  second  book  of  wliich  the  author  treats  '  de  arithmetica  &  militari 
geometria' ;  subsequent  editions  ib.,  1483  (two editions  this  year);  Paris, 
1483  (with  title  changed);  ib.,  1534,  fob  ;  ib.,  1555  (French  translation); 
(2)  'Speculum  Majus,'  by  Vincent  de  Beauvais  (Vincentius  Bellovacensis), 
the  greatest  mediaeval  encyclopaedia,  printed  at  Strasburg  by  Mentelin, 
i469(?)-i473,  10  vols.,  fol.  The  second  part  of  this  work  contains  one 
book  (no.  16)  on  mathematics,  in  which  is  given  a  brief  treatment  of 
algorism  (see  p.  5),  probably  the  first  wTitten  in  France  (c.  1250), 
although  M.  Henry  asserted  that  a  MS.  of  c.  1275,  which  he  edited, 
was  entitled  to  that  distinction.  Incomplete  editions  were  also  pub- 
hshed  at  Venice  in  1484,  1493-4,  1591.  The  early  editions  of  Priscian, 
'  De  figuris  et  nominibus  numerorum,'  are  mentioned  under  the  pub- 
lications of  1565.  (3)  The  Etvmologies  of  Isidorus,  Augsburg,  1472 
(p.  8). 

In  1480  an  anonymous  work  was  issued  from  the  Caxton  press  in 
London,  entitled  '  The  Mirrour  of  the  World  or  Thymage  of  the  same.' 
Chap.  I  o  of  this  work  began :  '  And  after  of  Arsmetrike  and  whereof  it 
proceedeth,'  and  this  was  probably  the  first  English  printed  matter  upon 
the  subject.  There  was  a  second  edition,  London,  1506,  and  a  third 
s.  1.  a.  (London,  1527  ?),  fol.  (See  also  Boethius,  p.  25,  and  Faber,  p. 
62.)  About  1480  there  was  published  at  Padua  a  folio  work  by  Richard 
Suiseth  (Suicetus,  Swincetus,  Swinshead,  Suineshevedus,  the  first  name 
possibly  Roger  or  Raymund),  entitled  'Opus  aureum  calculationum  per 
Johanem  de  Cipro  emendatum  et  explicit.',  with  subsequent  editions  at 
"Pavia  in  i488(?),  1497,  1498,  fob,  at  Venice  in  1505,  1520,  and  at 
Salamanca  in  1520.  I  have  seen  an  edition  of  this  work,  s.  a.,  assigned 
to  c.  1477.    For  Nicolaus  Cusa,  see  p.  42. 

GIORGIO  CHIARINI.     Ed.  pr.  1481.  Florence,  1481. 

A  Florentine  arithmetician  of  the  fifteenth  century. 

Title.  '  Qvesto  e  ellibro  che  //tracta  di  mercatantie  //et  vsan- 
ze  de  paesi.'    (F.  i,  r.    See  Fig.  5.) 

Colophon.  'Finite  ellibro  di  tvcti // ichostvmi  :  cambi  :  mone 
//te  :  pesi :  misvre  :  &  vsanze  //  di  lectere  di  cambi  :  &  ter//  mini 
di  decte  lectere  che//nepaesi  sicostvmaet  in //diverse  terre. 
Per  me  France //fco  di  Dino  di  lacopo  Kartolaio  Fiore//tino 
Adi  X  di  Dicembre  MCCCCLXXXI.  In  Firenze  Apreffo // 
almusiftero  di  Fuligno. '     (F.  102,  r.) 


PRINTKI)    HOOKS 


1 1 


Dcscriptio)t.  8°,  i  3.2  X  21.2  cni.,  ihc  text  bciiij;-  7.1  x  14. 1  iin. 
3  ff.  blank  +  6  unnuml). -f  c)r)  mimh.  (Roman  mimcials)  =  105  It., 
24  11.     Florence,   14S  1 . 

Editions.  Florence,  148 1,  8"  (here  described);  s.  a.  (1498), 
8°.  The  undated  edition  was  not  the  first ;  Coppin^er  has 
shown  that  it  was  printed  in  1498. 

While  this  is  not,  strictly  speaking,  an  arithmetic,  it  is  the  first  printed 
book  to  give  the  customs  relating  to  exchange  in  use  among  the  Floren- 
tine merchants  at  the  close  of  the  fifteenth  century.  It  is  the  source  from 
which  several  later  writers  drew  their  material,  and  is  ]jarticularlv  \alu- 

QVESTO  EELLIBRO  CHE 
TRACT  A  DI  MERCATANTie 
ETVSANZEDE  PASSU 

Fig.  5.   Title  of  chiarini's  work 

able  in  showing  the  nature  of  the  practical  jjroblems  of  the  time.  Copies 
of  this  first  edition  are  extremely  rare,  but  the  work  was  well  enough 
known  for  Paciuolo  to  appropriate  some  of  the  contents.  There  is  a 
(juestion  as  to  its  authorship. 

EUCLID.     Ed.  pr.  1482.  Ba.sel,  1562. 

Flourished  at  Alexandria  c.  300  b.c.    He  was  the  author  of  the  '  Elements,' 
the  basis  of  most  of  the  textbooks  on  geometry. 

Title.  '  Die  Sechs  Erftc  Bucher  //  Euclidis/  //  Vom  anfang 
oder  grund  //der  Geometrj.//  In  welchen  der  rechte  grund/nitt 
allain  der  Geometrj // (verfteh  alles  kunftlichen/gwifen/ vnd 
vortailigen  ge-//brauchs  des  Zirckels/  Linials  oder  Richtfcheittes 
vnd//andrer  werckzeiige/ fo  zu  allerlaj  abmeffen  dicnftlich) // 
fonder  auch  der  fiirnemftcn  ftuck  \nd  vortail//der  Rechcn- 
khunft/furgcfchriben  vnd // dargethon  ift.//  Aufs  Griechifcher 
fprach  in  die  Teiitfch  gebracht/aigene-//tlich  erklart/ Auch  mit 
verftentlichen  Exempeln/grund-//lichcn  Figuren/vnd  allerlaj 
den  nutz  fiirangen  ftellen-//den  Anhangen  geziert/  Derniaffen 
vormals  //in  Teiitfcher  fprach  nie  gefehcn  //worden.//  Alles  zu 
lieb  vnd  gebrauch  den  Kimftliebeiulen  Teiitfchen/fo   fvch  der 


12  KARA   ARITHMETICA 

Geo-//metrj  vnd  Rechenkunit  anmaflen/  mit  vilfaltiger  mlihe 
vnd  arbait//zum  trewlichften  erarnet/vnd  in  Truckh  ge-//geben/ 
Durch // Wilhelm  Holtzman/genant  Xylander/  //von  Aiigfpurg. 
//Getruckt  zu  Bafel.'     (P.  i.) 

Colophon.  '  Vollendet  durch  Jacob  Kiindig/  zu  Bafel/  in  // 
Joanns  Sporini  koften/im  jar  i562.//auff  den  dreyfzigften  tag 
des//Winmonats.'     (P.  199.) 

Description.  Fol.,  20  X  31.3  cm.,  the  text  being  12.5  X  25.7 
cm.  14  pp.  unnumb.  +  185  numb.  +  i  blank  =  200  pp.,  39-52  11. 
Basel,    1562. 

Editions.  The  editions  of  Euclid  have  not  been  considered 
in  this  work  except  in  so  far  as  they  relate  particularly  to 
arithmetic.  This  is  the  first  German  edition.  The  Plimpton 
library  contains  the  first  edition  of  Euclid  (Venice,  1482),  but 
since  this  has  no  arithmetical  work  except  Book  V  it  has  not 
been  included  in  the  list.  Several  manuscripts  of  Euclid  are, 
however,  included  in  the  second  part  of  this  bibliography  be- 
cause of  their  value  in  tracing  the  changes  in  the  forms  of  the 
numerals. 

This  edition  of  Euclid  is  mentioned  because  the  editor  has  thought 
it  necessary  to  add  to  Book  II  some  arithmetical  work.  In  particular 
he  gives  three  forms  of  multiplication,  first  from  left  to  right,  then  in  the 
usual  way,  and  finally  for  special  cases  in  which  the  short  processes  are 
involved.  He  also  considers  the  division  of  numbers  in  given  ratios, 
the  extraction  of  roots,  and  a  few  other  semi-algebraic  calculations. 

Euclid's  '  Elements '  contain  much  work  upon  the  Greek  theory  of 
numbers,  besides  what  appears  in  Book  V,  and  several  books  w^ere  pub- 
lished in  the  sixteenth  century,  embodying  this  material.  These  are 
mentioned  later. 

Otherworks of  1482.  In  1482  appeared  the  first  German  arithmetic, 
if  we  except  the  'Ars  Numerandi '  (p.  23).  It  was  written  by  Ulrich 
Wagner,  a  Niirnberg  Rechenmeister,  and  was  printed  by  Heinrich  Pet- 
zensteiner  at  Bamberg.  Only  nine  small  pieces  of  parchment  proof  sheets 
remain.  They  contain  the  following  colophon:  'Anno  diii  .  .  .  1482 
kri6.  lunij  p.  Henr.  peczensteiner  Babenberge  :  finit  Ulrich  wagner 
Rechemeister  zu  Niirnberg.'  '  Ludus  Arithmomachire,'  by  John  Sher- 
wood (Shirewode),  was  published  at  Rome  in  the  same  year.  (See  also 
p.  63,  and  on  Albert  of  Saxony  see  p.  9,  c.  1478.) 


PRINTED    BOOKS  13 

PROSDOCIMO  DE  BELDAMANDI,  and   LIVKRIUS. 
Ed.  pr.  1483.  Padua,  1483. 

Prosuocimo  dk  Hei.damandi  was  born  at  I'adua  c.  i370-t38o,  and  died 
in  1428.  He  was  educated  at  the  University  of  Tadiia,  and  also  taught  there. 
He  wrote  on  arithmetic,  music,  and  astronomy. 

JuHANNES  DE  LlVERius  (LiVERlls,  LiNERiis)  was  a  Sicilian  writer  on 
astronomy  who  flourished  c.  1300- 1350. 

Title.  '  Profdocimi  de  beldamandis  algo-//i'ifmi  tractatus 
perutilis  :  ncceffarius  //  foeliciter  incipit.  qui  de  generibus  cal-// 
culationum  fpecic  preteri.t  nullaj.q  falte // neceffaria  ad  h"*  art^ 
-"gnit65  fuerat.'     (F.  i,  r.    See  Fig.  6.) 

Colophon.  '  Algorifmus.  Profdocimi  de  beldamadis  //  vna  cum 
minuciis.  Johanis  de  liueriis.  hie //felicite'finit  Impreff'>  padue. 
Anno  .1.4.8.3 //die  zz.  februarii.'     (F.  21,  v.) 

Description.  4°,  15  x  20.8  cm.,  the  text  being  9X  14.7  cm. 
27  ff.  numb.,  32  11.     Padua,  1483. 

Editions.  Padua,  1483,  fol.  (here  described);  Venice,  1540, 
8°  (see  p.  15).  Boncompagni  could  learn  of  only  three  copies 
of  the  first  edition,  and  seven  of  the  second. 

This  rare  work  was  written  for  the  Latin  schools,  and  is  a  good 
exainple,  the  first  to  appear  in  print,  of  the  non-commercial  algorisms 
of  the  fifteenth  century.  It  follows  '  Bohectius  '  (Boethius)  in  defining 
number  and  in  considering  unity  as  not  itself  a  number,  as  is  seen  in 
the  facsimile  of  the  first  page.  Prosdocimo  then  treats  of  the  funda- 
mental operations  with  integers,  including  mediation  (division  by  2, 
which  the  author  places  before  duplation  or  multiplication  by  2),  pro- 
gressions, and  the  roots.  The  treatment  of  fractions  is  left  to  Liverius : 
'  Incipit  Algorismvs  de  mi-//nutijs  tarn  \'ulgaribus  quam  phyficis  magi- 
//flri  loannis  de  Liuerijs  Siculi.'  The  *  vulgar  minutes  '  were  the  com- 
mon fractions,  and  the  '  physical  '  were  the  sexagesimal  fractions. 
Towards  the  end  of  the  '  Algorismus  de  integris '  in  the  1540  edition, 
the  date  of  composition  appears  :  '  .  .  .  per  Profdocimum  de  Belda- 
mandis de  Padua  anno  domini  .1410.  die  .10.  lunij  compilata  fiifficiant.' 
(F.  D  5,  V.)  The  work  of  Prosdocimo  contains  the  first  reference  that 
I  have  seen  to  a  slate.  '  Indigebat  etiam  calculator  semper  alitjuo  lapide, 
vel  sibi  conformi  super  quo  scribere  atque  faciliter  delere  posset  figuras 
cum  quibus  operabatur  in  calculo  suo.'  (See  Fig.  6.)  It  is  probable, 
from  this  statement,  that  computers  of  his  time  actually  erased  the 
figures  in  the  galley  form  of  division  (see  the  Treviso  arithmetic,  p.  3), 


14  RARA   ARITHMETICA 

•3p)rofdodmi  dc  beldamaridie  algo^ 
nl'mi  iractdtae  pcruiilio  x  ncccffdnua 
focUcitcr  inctpicqui  dcgencnbue  cal^ 
colaaoimni  fpcde  p:etcn.t  nnlla^q  falic 
iiccdfaria  od  b^  art?  ^gnuoj  fiicral 

Tlncm  w%  plnnlnw  librw  aIgo:ifmf  nuaipa 
tJ0.rao0  circa  numcroe  ogandi  fane  uaru  e: 
atqj  ctoiofoe.g  licet  bom  c;«rtcTcf,atq3  vcri 
crattfi  faflidiofi:iu  ^pt  ipajj  rcgulojj  mnl'' 
litodmctu  ^  eap  dclcaioce^tu  ctia  ^ptcr 
ipap  opcratfono^jbafoeerv^Xbonc  fucrint  nd  nc.it  rat  t  tii 
am  ifti  modi  incm  fanidioft:  q;  It  in  abq? calculo  aflroloico  enci 
oti0yx^\<uhto7C  opato^  foam  a  caplremctpere  oponcban  da 
to  q?  crro:  fuae  adbuc  fatis  ^jpiqans  cjiftcrcf.  t  boc  jppi  ftgu 
rao  in  faa  opatoe  ddeta0»3nd^cbat  ctta  calcnlato:  femp  auq^ 
lapide  ud  fibi  ::>fo:muruj^  qno  fcnbere  atq^  fadlitcr  ddere  pct[^ 
figui  ad  cu  gbuf  opabat  in  calcnio  fuo.  iLt  9a  bee  eta  fane  fa;  1 
diofa  atq^  latjonofa  mibi  uifa  funtrd^fpofni  hbeU6  edere  in  quo 
cia  ilia  abicercn^:qai  etia  algonfmne  due  liber  dc  nomerie  de^ 
noiari  poterit.Sciao  tn  q^  in  boc  libdlo  poncf  nointcdo  nift  ca 
q  ad  calculu  necdfana  rontoUa  q  in  aliid  Itbrie  pracoce  anfme 
trice  tagunT.ad  calcnlii  nonecdfaria  ^jpf  b:enit3rejdiinucndo.  ^  . 
CrdoiA  ergo  Iibd!u0  ifle  dc  nnmene  traaare  bj:a  difiiiiitone  ^^^  A"^^ 
nueri  ipm  icboarc  odo.CTlijmeros  ergo  ^"^  ^ncltde.7?.fuc  ^^^liUf. 
i^eomctrie.  t f*  35obcctiij  pf fne  anfmetcc  fie  ditTfuiuTiiie  feo«+it*s 
ra0.  ell  maftitDdo  floe  qoatitae  difaeta  ex  nn  ranboe  ^iuktk  c 
tx  ttnftattboe  oggregattj^t  d:  nmfae  illadrq?  unaqneqj  rcecJt-  Vtm^m^A 
<if  una.  lp)CT  blc  g*  diffinitSj  nuert  b:cpofc0,quo?  unifas  no 
e  naem0,  IiQ  Ht  pncipiu  naeri^dato  q?  nuer^  et  :a  vocari  poUit  15 
go?  fnmcn'nae^X^)  cico  qno  rcaliqua  nuerarc  pofTumtig*  et 
fiiolargo?  occqw^  niicr^  in  ^cdTa  bniufi  libdii.  in  q?  itik  vnitaf 

Fig.  6.    First  page  ok  prosdoclmo  de  beldamandi 

as  the  Hindus  had  done  on  their  sand  or  dust  abacus,  instead  of  cancel- 
ing them  in  the  manner  explained  in  the  early  printed  arithmetics. 

The  best  discussion  of  the  lives  and  works  of  Beldamandi  and  Liverius 
is  in  the  Boncompagni  BiiUetino,  vol.  XII. 


PRINTED   BOOKS  15 

Other  works  of  14SJ.  \';illuriiis  (j).  10).  In  this  year  the  second 
(rerman  arithmetic  was  jirinted  at  Bamberg,  only  one  (inc  omjjlete)  copy 
being  known.  It  was  possibly  written  by  Ulrich  Wagner  (p.  12),  and, 
like  the  1482  work,  it  was  printed  by  Petzensteiner,  as  appears  from 
the  following  colo]jhon  :  '  In  zale  Xpi  .1483.  kl '  .17.  des  Meyen  Rech- 
nung//in  mancherley  weys  in  l^abenberg  durch  henr^ // jietzenfteiner 
begriffen  :   volendel.' 

PROSDOCIMO   DE  BELDAMANDI,  and   LIVKRIUS. 
Ed.  pr.   1483.  Venice,  1540. 

See  p.  13. 

Tith-.  '  Algorismvs  dc  In//tet;ris  Magistri  Prosdoci-//mi 
Debeldamandis  Pataui  iimul  cu  algorifmo  dc  do-//minutijs  feii 
fractionibus  magiftri  loanis  de  Liuerij  //  ficuli.  Rcinlegratus  ab 
erroribus  comiflis  a  fcri-//ptoribus,  a  me  Federico  Delphino  ar- 
tium,  &// medicine  doctore,  mathematicarum  difci//plinaru  in 
celeberrimo  gymnafio  Pata//uino  publico  profefforc,  additis  ali// 
quibus  verbis,  in  aliquibus  locis,//  pro  maiori  claritate.  Et  da-// 
tiis  impreffioni  ad  infta-//tiam  meoR  fcholariii // nunc  algorifmu 
// maxime  de-//^ideran-//tium. //  Venetijs.  M.  D.  XXXX.' 
(F.  i,r.) 

Colophon.  '  Venetijs  per  loannem  Antonium  de  Vulpinis  de 
Ca-//ftrogiufredo.  Anno  domini  .M.  D.  XXXX. //tlie  octauo 
menfis  Aprilis.'     (F.  52,  v.) 

Description.  8°,  9.6  X  14.5  cm.,  the  ])rinted  part  being  7.  i  X 
12.5  cm.  44  ff.  unnumb.,  30  11.  Venice,  1540.  (This  particular 
copy  has  8  ff.  more,  the  sheet  F  appearing  in  duplicate.) 

See  p.  13. 

RAPHAEL  FRANCISCUS. 

Ed.  pr.  1484.  S.  1.  a.  (F"lorence,  c.  1516). 

Raffaelk  Francesco.    A  Florentine  philosopher  of  the  latter  part  of 
the  fifteenth  century. 

Title.  'Verificatio  Vniversalis  in  // regulas  Ariftotelis  de 
motu  non  rece-//clens  a  comuni  Mathema-//tico:p  doctrina.' 
(F.  I,  r.     See  Fig.  7.) 


i6  RARA    ARITHMETICA 

Description.  4°,  13  X  19.9  cm.,  the  text  being  9  x  16.7  cm. 
8  ff.  unnumb.,  41-42  11.    S.  1.  a.  (Florence,  B.  Zucchetta,  c.  15  16). 

Editions.  Pisa,  1484,  8°;  this  edition,  s.  1.  a.  (Florence,  c. 
1516). 

This  is  a  brief  treatment  of  proportion,  hardly  worthy  of  ranking  as 
an  arithmetic.    The  applications  relate  to  problems  of  Aristotle. 

VERIFICATIO  VNIVERSALIS  IN 

regulas  Aridorelis  de  mom  noti  rece^ 

dens  a  comuni  Mathema" 

tico3«doclrina^ 

Fig.  7.    Title  of  the  verificatio  of  franciscus 
PIETRO    BORGHI.     Ed.  pr.  1484.  Venice,  1484. 

PlERO  BoRGl.    A  Venetian  arithmetician  ;  died  after  1494. 

Title.  '  Qui  comenza  la  nobel  opera  de  //  arithmethica  ne  la 
qual  fe  tracta  //  tute  coffe  amercantia  pertinente  //  facta  j  compi- 
lata  p  Piero  borgi // da  veniefia.'  (F.  2,  numbered  i,  r.  See 
Fig.  8  for  the  first  folio.) 

Colophon.  'Nela  inclita  cita  de  venetia  a  9orni  .2.//augufto 
.1484.  fu  impofto  fine  ala  pre-//fente  opera.'  (F.  1 18,  v.) 

Description.  4°,  14.3  X  19.3  cm.,  the  text  being  8.4  X  13.4  cm. 
2  ff.  unnumb.  +  116  numb.  =  118  ff.,  37-38  11.    Venice,  1484. 

Editions.  Venice,  1484,  4°  (here  described)  ;  ib.,  s.  a.,  which 
Riccardi  thought  might  be  earlier  than  1484,  since  Ratdolt,  the 
printer,  published  books  in  Venice  as  early  as  1476;  ib.,  1488, 
4°  (see  p.  19)  ;  ib.,  1491,  4°  (see  p.  20);  ib.,  1501,  4°;  ib.,  1505; 
ib.,  1509,4°;  ib.,  1 5  17,  4°  (see  p.  20);  ib.,  1528,  4°  (see  p.  21); 
ib.,  1534,  4°  (see  p.  21);  ib.,  1540,  4°  (see  p.  21);  ib.,  1550,  4° 
(see  p.  22);  ib.,  1551  ;  ib.,  1560 ;  ib.,  1561,  4° ;  ib.,  1567  ;  ib., 
1577.  It  was  at  one  time  thought  on  the  testimony  of  Maittaire 
that  there  was  an  edition  of  1482,  but  it  has  been  shown  by  sev- 
eral bibhographers  that  the  first  edition  is  that  of  1484. 

This  is  the  very  rare  first  edition  of  Borghi's  treatise,  the  second 
commercial  arithmetic  printed  in  Italy  and  long  thought  to  be  the  first. 
Xhis  particular  copy  belonged  to  Cotmt  Paolo  Vimercati-Sozzi. 


PRINTin)    BOOKS  17 

The  text  of  the  first  edition  is  closely  followed  in  that  of  14S8  (see 
p.  19),  except  for  the  index,  'Tauola  de  li  capitoli  ;Dtegnudi  i  qfla 
opa'  (f.  118,  r.).  This  does  not  appear  in  the  second  edition,  at  least 
in  Mr.  Plimpton's  copy.  The  letters  S  H  S  U  which  ajjpear  twice  are 
thought  to  stand  for  J  H  S  U,  Jesus,  possibly  changed  on  account  of 
some  conjectured  pronunciation.  They  appear  the  second  time  on 
f.  118,  v.,  in  connection  with  a  set  of  verses  beginning  as  follows  : 

'S  H  S  U 

Quanto  latua  memoria  ct  alto  in<jegno 
vaglia  lie  huitlimetica  hai  iiioftrato 
nel  prefente  volume  compilato 
petro  borgo  date  veneto  degno.' 

In  the  verses  appears  the  name  of  the  printer  : 

'  Ma  limprelTor  de  augufta  Krrardo  experto 

di  lopera  prefente  ftampatore 

degno  e  non  di  mediocre  laude  certo.' 

This  folio,  as  already  stated,  does  not  appear  in  the  Plimpton  copy  of 
the  1488  edition,  and  since  the  later  printer  was  not  the  same  it  prob- 
ably never  appeared  after  1484. 

This  work  is  more  elaborate  than  the  Treviso  arithmetic,  and  had 
far  greater  influence  on  education.  More  than  any  other  book  it  set 
a  standard  for  the  arithmetics  of  the  succeeding  century,  and  none  of 
the  early  textbooks  deserves  more  careful  study.  Borghi  first  treats  of 
notation  (see  Fig.  9),  carrying  his  numbers  as  high  as  '  numero  de  mil- 
lion de  million  de  miUion,'  and  making  no  mention  whatever  of  the 
Roman  numerals.  In  the  same  spirit  he  eliminates  all  of  the  mediaeval 
theory  of  numbers,  asserting  that  he  does  this  because  he  is  preparing  a 
practical  book  for  the  use  of  merchants.  '  VA  nota  che  fono  //  niieri  de 
piui  maniere  fi  cho-//me  dichiara  Boetio  in  el  fuo  //de  arithmetiche  Ma 
volen-//do  hora  tratar  de  quelle  chof//fe  che  folo  amerchadati  apertien : 
pero  tratado  folo  de  quelli//che  ale  choffe  merchadatefche  fono  necef- 
iarij  io  laffero  ogni  //  altra  maniera  de  numeri.'    (F.  numb,  i ,  v.) 

The  sequence  is  now  peculiar,  for  multiplication  is  the  first  operation 
treated.  (' CChe  coffa  fia  moltiplichar,'  f.  numb.  7,  r.)  First  comes 
the  table,  arranged  in  the  column  form,  unlike  the  Boethian  type  of 
arithmetic,  which  preferred  the  square  array.  In  addition  to  the  prod- 
ucts through  '9  uia  10  fa  90,'  the  products  of  12,  16,  20,  24,  32,  36, 
by  2  ...  10,  are  given,  these  having  been  necessary  on  account  of  the 
monetary  tables  of  the  time.    The  author  then  gives   the  method  of 


i8  RARA   ARITHMETICA 

checking  by  casting  out  y's  and  9's.  (' CDela  pruoua  del  .7.',  '  Dela 
pruoua  del  .9.',  f.  numb.  8.)  Then  follows  multiplication  '  per  colonna ' 
(i.  e.,  by  reference  to  the  columns  of  the  table,  '  CDel  multiplicar  p 
cholona,'  f.  numb.  9,  r.),  with  its  checks  by  7  and  9,  and  '  per  crocetta ' 
(our  '  cross  multiplication,'  '  CDel  multiplicar  per  chroxeta,'  f.  numb. 
13,  v.),  showing  that  these  were  the  common  methods  in  Venice.  Divi- 
^.  ,  ...  sion  is  then  explained  by  the  gal- 

Cbi  w aitematcmaticbeba  piacere     i„^ . „^„^  /<  ^  cnmo  fe  die  nartir  n 

£bCtCfl50n&icmC5aclp:imOSradO        V?    I-  ^r  die  partir  p 

Buanticbc  ci quelle  tenti  el  vado        ^^^^^°  ^^^  ^^^^^°  P'"'^^"^'  ^-  "^"^^b- 

©00(1  lo  p:erentc opera  vedere  ^o,  r.),  our  present  method,  then 

Perqudtaluipotracertofapcre  known  as  the  method  of  giving, 

SeerrozfamnclCQlculonotadO  'a    danda,'    and    described    by 

gcrqucftadTcrpOtracerflflCadO  paciuolo  and   Calandri   (pp.   54 

Ufo2mdrcomDttmo manure  ,      x      ^  ,  ^i-     ^ 

amercbadanritnoftavtilitade  ^^^  ^y),  not  bemg  mentioned. 

^^aro  la  prcfenre  opera  cafatozi  ^^^^  ^^^^^'^  addition  (although 

^arai'nfar^O.>r:'grat?f<<Cl!rt'^de  this  was  used  in  mukiplication), 

prrqucftavederantuttliicrroji  subtraction,    denominate    num- 

^dCiquatcrmfoilavCritadC  .      bers,     common    fractions     (also 

feanariacqu.neranoesrandiboi.o2i     beginning  with  multiplication), 

5"lapatriaebefuoei      ,     .  ,u       /.  t^   1     •      1    ji 

0ap2an  fer  lerafon  i>etutte  gente        '""^^  °^  ^^'^^  (  ^^  ^^  '"^^g^^^  ^^^ 

Perleflgurecbe Ton qni'oepcntC  •3-')'  partnership,  barter,  alhga- 

FiG.  8.   First  page  of  the  first  edi-    ^ion,  and  false  position.  The  rule 

TioN  OF  BORGHi's  ARITHMETIC  ^f    three    had    been    developed 

many  centuries  earlier  by  Orien- 
tal arithmeticians.  It  was  one  of  the  inheritances  from  the  Arabs,  and 
was  not  improbably  learned  by  the  Venetian  traders  through  their  con- 
tact with  the  East.  Partnership  was  to  the  fifteenth  what  the  corporation 
is  to  the  twentieth  century,  and  it  is  only  very  recently  that  "  partner- 
ship involving  time"  was  thought  to  be  a  necessary  subject  of  study. 
Barter,  a  subject  until  fifty  years  ago  common  in  American  textbooks, 
was  necessary  at  a  time  when  currency  was  not  so  plentiful  as  now. 
Alligation  was  a  practical  topic  in  connection  with  the  coinage  of 
money  in  the  days  when  minting  was  not  the  monopoly  of  great  cen- 
tralized governments.  The  rule  of  false  position  was  an  Oriental  device 
which  we  have  now  replaced  by  the  equation  ;  it  was  found  in  school- 
books  in  various  countries  until  the  second  half  of  the  nineteenth  cen- 
tury. The  problems  are  generally  practical  for  the  time,  and  they  reveal 
some  interesting  facts  concerning  business  customs  at  the  close  of  the 
fifteenth  century. 

Of/ier  works  0/1484.    Albert  of  Saxony,  p.  9,  c.   1478;   Vincent 
de  Beauvais,  p.  10. 


PRINTIH)    HOOKS 

Ktl.  pr.   1484. 


19 

Venice,  148S. 


1000609 


I  lOOOOi 


1 1  i  0000 


PIETRO   BORGHI 

See  p.  16. 
TttU.    '  Chi  de  arte  malematiche  ha  piacerc // Che  teiif^on  di 
certeza  el  primo  grado // Auanti  che  di  quelle  tenli  el  vado// 
Vogli    la    prefente  opera  vedere // Per    quefta   lui  ])()lra  certo 

CJComo  fi  fo:mflno  miUon 

_^..::-=s  H  million  flclo«d?a  fc  bi'c  foi wor  per Mtc  figure 
fil  So^M  »"<1"^^'>  "^^^^  •  'oooooo  .pcrtbc  hfcpcima 
'llKwa  f^S"^^  tic«  cIlMOfio  Dcnitara  ocimara;:cpcrcbe 
B^^^'K  tnillcmiara  .fanovnotmllion:ctcfrcijdoii)<iucl 
1!  E^M|  luogo  lafikjura  cbcr/pKuijca  vuo  pcro  bene 
l|||ge^l|  cditovnonMlic»n.^a.fqucftpmodo.  1  100000 

^^^ "  ufna'vnonjili'of)  ccctjto  miliaijpcrcbeoltrad 

itjilli'ofitm  hOQO  tccentcmr&  oc  mura'.rouo  lahijura  cbc  ripzc 
jcctitavfjoficbcbciiccditovno tnt'Kton cccnto  nn'(ia.3K)a.fti 
qocflo  modo.  1  s  loooo.oina  vrjoinihoncccijtoc  dicjccmilia 
ocrchcoUuclmilionccmto  fm'Iia  :m  luogo ocic  oejcciieoc^ 
luiarfono  lafi'sura  cbc  rip:e]ccnt9  vno-.A  cbc  benccdi'to  viio 
ttiidoneccntocdicircttiilia.Z&aiiiqtKdoinodo.  I  1 1  looo.Dt' 
riavnmiiioti  cento  cvndocciiu'lfa  per  cbc  oltraclmilion  cento 
edicicc  mili&'in  luogo  be  nuincn'oemfarfonolafigura  cbc  rip 
wnta  vno:ft  cbc  bene  cdi  to  vii  tnilion  caj  to  c  vndcice  milia.^a 
in<iucftomodo.  1 1  1 1  loo.biria  vnmi'Iion  cento  evndeceint^ 
Hi  e  ccntorpcrcbc  oltra  elnitlion  cento  e  vndcirc  milia:  in  luogo 
Occffimplicccentenarfono  lafiguracbe  ri'pzaenta  vnoifube 
bciic  cditovnmilion  cento  c  vndcice  nnlia  c  ccuto.^a  inqucfto 
tno^o.  I  i  1 1 1  lo.einavnnnljon,  cento c  vndcrcnnlla  cento 
£dic]cc:pcrcbc  oltra  clmilion  caitocvndcicc  milio  e  cento :  i 
luogo  belefimpUceocrene-.fono  lafigoracbe  n'pzejcentavno. 
:^ainciucftomodo.  1 1  1 1  1 1  i.biriavnnu  lion  cento  evndc 
re  mi'Iiacentoc  vndcre.pcr  cbe  ancbe  m  (uogo  ocle  fiinplicc 
vnita  Xotto  lafigura  cbcr/p^erenta  vno.ficbe  bene  cditovn' 
fnilioncentoe  vndcjtemilta  cento  c  vndejce.ctcbori  p:occdcdo 
perfina.<?^99^^<?.poncndo  fcinpjc a ifuoliiogiquefc figure 
rcpzcjccntante  quelinunicri  oucrooerenc  occncenara .cbe fi' 
nominaetcetcra.cqucftobafta  cercba  loamfliftranicntoocl 
mimerar.bencbeininfini'tumfipoJin  p:cceder.mo  cbontvna 
jcncral  figura  mi'fojcero  oicbiarirquantopotcfTeacbadcr.cc 
fgranoqucflo  fottopoftd 

Fig.  9.    From  bok(;hi's  arithmetic,   1488   kuition 

fapere  //  Se  error  fara  nel  calculo  notado  //  Per  quefta  cffer 
potra  certificado // A  formar  conti  di  tutto  maniere/'/A  mer- 
chadanti  molta  vtilitade  //  Fara  la  prefente  opera  e  afatori  // 
Dara  in  far  conti  gran  facilitade  //  Per  quefta  vederan  tutti  li 
errori // Ede  iquaterni  foi  la  veritade // Danari   acquifterano  e 


I  I  1 000 


I  I  M  1  09 


I  I  I  I  I   0 


I  I  I  I  I  I  I 


9^9^09^ 


20  KARA   ARITHMETICA 

grandi  honori//In  la  patria  e  de  fuori // Sapran  far  le  rafon  de 
tutte  gente//Per  le  figure  che  fon  qui  depente.'    (F.  i,  r.) 

'  Qui  comeza  la  nobel  opera  de  //  arithmeticha  ne  laqual  fe 
tracta//tute  coffe  amercantia  pertinen-//te  facta  2  compilata 
per  Piero//borgi  da  Veniefia.'    (F.  2,  r.) 

Colophon.  '  Stampito  in  Veniexia  per  zouane  de  Hall'  1488.' 
(F.  95,  V.) 

Description.  4°,  14.8  X  20.7  cm.,  the  text  being  12.4  x  14.9 
cm.    95  ff.,  44  11.     Venice,  1488. 

Editions.  See  p.  16.  This  is  the  third  edition,  and  is  nearly 
as  rare  as  the  first.  Proctor  mentions  three  books  from  the 
press  of  the  printer,  John  Leoviller,  of  Hall  (Halle  1). 

The  text  is  practically  verbatim  with  that  of  the  first  edition  (p.  16). 

PIETRO   BORGHI.     Ed.  pr.  1484.  Venice,  1491. 

See  p.  16. 

Title.  The  title  page  is  the  same  as  that  of  1488,  except  for 
the  letters  SHSU  which  precede  the  former. 

Colophon.  'Nela  inclita  citade  venetia  a  zorni  .22.//ottubero 
.1491.  u  impofto  fine  ala  pre//fente  opera.// Libro  dabacho.' 
(F.  100,  V.) 

Description.  4°,  15.4  X  20.8  cm.,  the  text  being  11.7  X  16.8 
cm.    100  ff.  unnumb.,  40-43  11.    Venice,  149 1. 

See  p.  1 6.    This  is  the  fourth  edition. 

PIETRO   BORGHI.     Ed.  pr.  1484.  Venice,  1517. 

See  p.  16. 

Title.  This  is  substantially  the  same  as  that  of  the  1484 
edition  already  described. 

Colophon.  '  CStampata  in  Venetia  per  lacomo  pentio  da 
Lecho  ad  infta//tia  de  Marchio  Seffa  &  Piero  di  Rauani 
compagni  //anno  dfii  .15  17.  adi  .25.  de  zugno.'     (F.  100,  v.) 

Description.  8°,  15.5  X  21.6  cm.,  the  text  being  13.3  X  17  cm. 
2  ff.  blank  +  100  numb.  =  102  ff.,  41  11.    Venice,  15  17 

See  p.  16. 


PRINTED    HOOKS  21 

PIETRO   BORGHI.     Ed.  pr.  1484.  Venice,  1528. 

See  p.  16. 

Title.  This  is  substantially  the  same  as  thai  of  the  1484 
edition  already  described. 

Colophon.  '  CStampato  in  Venetia  per  Fracefco  liindoni,  & 
Mapheo // Pafyni  compagni.  Nel  anno  . M.D. XXVIII. // Adi 
.XVIII.  Del  mefe  di  Zenaro.'    (F.  100,  v.) 

Description.  8°,  15.1  X  20.5  cm.,  the  te.xt  being  13.4X  17.1 
cm.    100  ff.  numb.,  41  11.     Venice,  1528. 

See  p.  16. 

PIETRO   BORGHI.     Ed.  pr.  1484.  Venice,  1534. 

See  p.  16. 

Title.  This  is  substantially  the  same  as  that  of  the  1484 
edition  already  described. 

Colophon.  '  CStampato  in  Vinegia  per  Fracefco  Bindoni,  & 
Mapheo // Pafini  compagni.  Nel  anno  .M.D. XXXIIII.// Adi 
.25.  Del  mefe  di  Settembre.'     (F.  100,  v.) 

Description.  8°,  15.4X  20.9  cm.,  the  te.xt  being  13.5  x  17.1 
cm.    100  ff.  numb.,  41  11.     Venice,  1534. 

See  p.  16.    The  various  editions  changed  but  little. 

PIETRO   BORGHI.     Ed.  pr.  1484.  Venice,  1540. 

See  p.  16. 
Title.  '  Pietro  Borgo  //  Libro  de  Abacho.//  Chi  d'arte  Mathe- 
matice  ha  piacere//Che  tengon  di  certezza  il  primo  grado // 
Auanti  che  di  quelle  tenti  il  vado//Vogli  la  prefente  opera 
vedere.//  Per  quefta  lui  potra  certo  fapere  //  Se  error  fara  nel 
calculo  notado  //  Per  quefta  effer  potra  certificado  //  A  formar 
conti  di  tutte  maniere.//  A  merchadanti  molta  utilitade  //  Fara 
la  prefente  opera  e  a  fattori  //  Dara  in  far  conti  gran  felicitade 
//  Per  quefta  uederan  tutti  gli  errori  //  E  delli  quaterni  fuoi  la 
ueritade  //  Danari  acquiftaranno,  e  grandi  honori.//  In  la  patria  e 
di  fuori  //  Sapran  far  le  raggion  de  tutte  gente  //  Per  le  figure  che 
fon  qui  depente.// Auenga  che  alquanto  jier  me  fu  promeffo  affai 


22  RARA   ARITHMETICA 

fufficientemente // alia  promeffa  fatisfaceffe,  niente  dimancho 
per  fatisfar  alle  pre-//giere  di  qualch'uno,  e  maffime  di  alcuni 
Impreffori,  iquali  era-//no  per  ftampar  la  prefente  Opera,  ho 
uoluto  alquanto  ampliar  la  di  qualche  gentilezza  oltra  quello  che 
prima  pmiffe,  benche  //  di  quello  che  fe  potria  dir,  quefto  fia  vna 
minima  parte,  pero  cli  //chi  uoleffe  metter  pur  la  centifmia  parte 
di  quello  che  fi  potria // poner,  el  faria  molto  piu  la  gionta  di 
quello  che  fia  tutta  I'opera // infi.  Et  pero  pro  nunc  mi  paffo 
con  alcune  cofette  aggionte  nel  //  ligar  de  metalli,  lequal  princi- 
piano  a  carte  .JJ.  &  anchora  in  fin//de  I'opera  con  dieci  cafi 
affai  piaceuoli  &  leggiadri  comincian//do  a  carte  .98.  da  quello 
che  dice.  Le  vno  che  compra  tre  pezze//de  panno  per  ducati 
.70.  &c.  Et  fe  le  mente  di  quelli,  iquali  me//hanno  pregato 
non  f uffmo  a  fuo  modo  fatisfatte,  prego  quelli  //  me  habbino 
per  ifcufato.'     (F.  i,  r.) 

Colophon.  '  Stampato  in  Venetia  per  Bernardino  de  Bindoni.// 
Ne  I'anno  .M.D.XL.    Del  mefe  di  Ottober.'     (F.  100,  v.) 

Description.  4°,  15.5  x  20.9  cm.,  the  text  being  13.2  x  17  cm. 
2  ff.  unnumb.  +  98  numb.  =  100  ff.,  39-40  11.     Venice,  1540. 

See  p.  16.  This  is  the  eleventh  edition.  The  title  is  considerably 
extended  and  some  changes  are  made  in  the  text,  chiefly  in  the  way  of 
added  matter. 

PIETRO   BORGHI.     Ed.  pr.  1484.  Venice,  1550. 

See  p.  16. 

Title.  The  title  page  is  substantially  identical  with  that  of 
the  1540  edition  already  described.  It  bears  the  date,  'Anno 
Domini  M.  D.  L.' 

Colophon.  '  CStampato  in  Vinegia  per  Francefco  Bindoni, 
&//Mapheo  Pafmi.  Nell  Anno  .MDL.//Adi  .21.  Del  mefe 
di  Nouembrio.'     (F.  100,  v.) 

Description.  4°,  15.6  x  20.9  cm.,  the  text  being  12.9  x  17.9 
cm.     100  ff.  numb.,  41  11.     Venice,  1550. 

See  p.  16.  This  is  the  twelfth  edition,  and  at  least  five  subsequent 
editions  appeared  in  the  sixteenth  century. 


PRINTED    HOOKS  23 

ANONYMOUS.     Ed.  j^r.  c.  1485.  S.  1.  a.     (Cologne  ?,  c.  1485). 

Title.  '  Ars  numerandi.  Incipit  copendiofus  tractatul^'  quin 
//tupliciu  dclonu  numealiu  in  quo  docet'//luculet'  quo  ordiant' 
variatur  c>ponut'//et  abinuicem  deriuatur  dictones  nume//  rales.' 
(F.  2,  r.  ;  see  Fig.  10.)  Without  abbreviations  this  would  appear 
as  follows  :  '  Incipit  compendiosus  tractatulus  quintuplicium  dic- 
tionum  numeralium  in  quo  docetur  luculenter  quomodo  ordinan- 
tur  variantur  componuntur  et  ab  invicem  derivantur  dictioncs 
numerales.' 

Description.  4°,  14  X  20.6  cm.,  the  text  being  10.8  X  15.2  cm. 
6  ff.  unnumb.  (i  blank),  35  11.    S.  1.  a.  (Cologne  .?,  c.  1470- 148 5). 

Editions.  There  was  no  other  edition.  The  date  of  this  rare 
book  is  uncertain.  The  style  of  type  has  led  to  the  assertion 
that  it  was  printed  in  Mainz  by  Fust  and  Schoeffer,  about  1470. 
The  book  is  not  mentioned  by  Hain,  nor  is  any  copy  known  in 
the  French  libraries.  Coppinger  believes  that  it  was  printed 
in  Cologne  by  Ulrich  Zell  in  1485,  and  in  this  he  is  followed  by 
Zell's  biographer,  Merlo.  The  British  Museum  catalogue  gives 
this  date  and  printer,  but  questions  each.  A  comparison  of  the 
water-marks  in  this  copy  with  those  of  the  fifteenth  century 
which  are  described  in  standard  treatises  (e.  g.,  Sotheby,  E.  L., 
Primipia  Tjpograp/iica,  London,  1858,  vol.  Ill)  fails  to  throw 
any  light  upon  the  date.  Riccardi  attributes  it  to  Zell,  c.  147 1, 
who  had  been  an  apprentice  of  Guttenberg,  but  had  left  Mainz 
at  the  sacking  of  the  city  in  1462. 

The  book  is  not  strictly  speaking  an  arithmetic,  but  a  treatise  on 
grammatical  usage  as  applied  to  numbers.  A  considerable  portion  of 
the  text  is  occupied  with  the  distinction  between  ordinals  and  cardinals, 
and  the  methods  of  using  them. 

Other  works  0/1485-1487.  Albert  of  Saxony,  1487,  p.  9  ;  c.  1478. 
In  1485  there  was  published  at  Bologna,  edited  by  Pietro  Almadiano  of 
Viterbo,  a  quarto  work  by  Nicolo  de  Orbelli  (Nicolaus  Orbellis)  entitled 
'Compendium  considerationis  matematice  quo  ad  aritmeticam  et  geome- 
triam  sunt  necessaria.'  Orbilli's  'Cursus  librorum  philosophic  naturalis,' 
which  appeared  in  1494,  4°,  and  at  Basel  in  1503,  4°,  contained  2  pp. 
on  arithmetic.    See  Isidorus,  p.  8,  1483. 


24 


RARA    ARITHMETICA 


Ard  nummnbi  * 

Inapit  copenbiodis  tmdatul^qmn 
tupitdu  t)aonu  numcaltu  m  quo  tocc^ 
luculct' quo  <»t)iaii^vatiatur  oponii^ 
ttabmuicQm  t)6riuitur  bidonca  numc 

Jf^ottu  numcnod  mtpJttSttu  qbat>icu 

^      turcarbmatcff«qfcop5t«dics*qtat)iftrt 

butiuc  fiuc  &i4>tific*qfcBcwt>talc6»a  qta 

muldpticatiuc  (tuc  at^ubtalcff '(Ht  itota 

t)n  ^  numcti  t>iucc(t0  toidJiortib?  figiicaf  a  cjc 

varia  fi§^nt>i  manccic  cotrabut  mtr  fc  bintia  ct 

t)iu(ttsi(^  ut  patcb^  tPcbisaut^dorrib^poi 

t>m$  cf>t)i(»t)U'*<£t  ptio  t«  carbTalifcsq  t>tcutur 

io  cai:t)talc9«  cp  ficut  oOiu  -^tif  ^ca  carbmc-ct 

tnnittf  <ij  ita  baonc*  aUc  ndaUff  -f  tutwt  a  «?^ 

)>ttcatar  cca  iOas  'Vel  Dicuf  cattmalcs  q(t  pn 

cipalc^  '  cf  tfconce  ahc  nualc?  ab  ifh^  bnt  oii 

^ncVcl  t)icu^  qft  ptidpaU^  cp  pticipafe  fig^nt 

nii;     <£t  fcicdu  ^  t)upHccff  flit  t)c5cs  cardma 

icffjqt>a'(^c5crct3C  ct  qucda  abftf^«  tDc  cocrctC 

aut  cjaromaUbscp  ts  ipis  atic  tKtiuan^Pimio  c 

t)tf m  que t>c6ncs  dcpcodutnumcmdo    '^nue 

t>uo^tt»3'qt»attuc«'qncp- (cjC'fcpte*  o  Ao'ttouc 

fcece'VntKd't>uot«d'ti«t«d'qt4attuattcd»  qtttsd 

fc&ed'f  (cfdcd  *  a nuti^  ft^rtcd  p  >>  tcccfcp tc* a 

non  tebj  mfrcrpoi  l:cc  itu<*io  ct  it  fint  feuc  tjcocf 

t»xoOx>  »tK«»noucm*  Vi  0tnti  *  vi  gwtxun?"  vi  gi 

tit>uo«afi<:urqj  at>tn0mta(uo  m<?do-''2^ugm 

ta'tugmtaun^'trigmtatouo'  a  fie  (uo  mode  t« 

aljjs^cTXuadtagmta  (TluTquagmta'Sc^tragrta 

Scptuagmtti'CD<*ogmta::fl7ona0rtnta.crcntu» 

ecntuun?'c5tuduo»  a  fie  (iio  mo  ufqj  t)U(XtoS' 

^u(iti'tC'ta'buc«tiun?'t>ucvCidito«  a*^  vlt«u? 


(Ttuaw^ma 
niimctalcs 

car&maU0« 


^uptidaftf 
<:art>matia' 


J^  eonar«ta 
tioianuatia 
fie  numcta* 


Fig. 


FlKSl"    PACE    OK    THE    ANONYMOUS    ARS    NUMERANDI 


PRINTKI)    HOOKS 


25 


ANICIUS  MANLIUS  SEVKRINUS  HOKTIIIUS. 

Ed.  pr.  1488.  Augsburg,  1488. 

BoETlus.    Horn  at  Rome  c.  480;  died  tliere  October  25,  524.    He  was  a 
Roman  senator,  a  philosopher,  and  the  last  of  the  great  Latin  writers. 

Title.    '  Arithmetica  boetij.'    (F.  i,r.) 

Colophon.  '  Finit  arithmetica  Boetij  bene  re//uifa  ac  ftdeli 
ftudio  emendata  Im//preffa  per  Erhardu  ratdolt  viri  fo-// 
lertiffimi    eximia   iduftria  1  mira    im-//primedi   arte :    qua   nup 


Jrtcipiunt&iiolibiiOc  ^Iritbmctt/ 
ca  anitij  manili;  fcumiii  boetij  vi/ 
ri  clarKfimi  x  illivftnlTimi  c  jc  cofulis : 
o:^inari):patricv  :a&  patriciiitnfim 
macbnm. 

TlOan^isaccipi 
enMfquemiineri 
bnsitarcctcoffu 
ciap:ccipuc  inter 
cos  gfofcmagni 
faciunt  cftimant 
fiUqui&ooftabit 
nccabboc  alius 
qd  libcralius  affcrrct  iniiciuii :  ncc 
ab  iUo  vnqi  qd  iiicitnSiiis  bcniuolc 
tia  coplecteret  accepni.'Df  c  ip  fc  co 
fi&erans :  atttiU  non  tgnaiia  opiim 
po&eraquibusa&faamis  nibil  in 
ftracriUQ  eft;  oi  baben&i  (Ttis  mean 
^lut:a&t^mtunlbllvllillsciica  fi' 
biviao:anim'calc3tafiibiccit:fc0 
ca  qii?  ZT  ga'carii  opiilenria  littera 
re  in  rotnanc  ojationis  tb^aunttn 
(upta  couejcirK'  .^ta  cni  md  quocg 
opcris  mibi  ratio  coftabit  :ft  que  cc 
fapientif  Soctrinis  elicniifapicntiP 
fimi  iubicio  copzobent .  ai&ea  igit 
vctam  magni  labozis  cffectus  niu? 
tanm  cjcpeact  ejtramc:  nee  in  aurea 
^»irc  publicas  nifi  &oct?  fcntctic  a 
IlipiUatione  mtat.3In  quo  nibil  mi 
rilvi&eri  &cbet:cii  iJ)  opus  qd  fapic 
tic  inuentaperfequit:  non  auaozis 
f5  a'ieno  inciibit  arbitrio.Suis  gp/ 
pc  inftrumenris  res  ratioitis  ejrpcn 
J)imr:cii  iubiciitm  cogit  fubtrc  piu^ 
t>cntis.Seb  buic  munufculo.no  ea 
^e^n  que  ceteris  imminent  artibua 
mwnimcntacoftxtuo  .Tlequc  enim 


fere  Vila  fie  ciiais  abfoliita  partib'  nui. 


a  fciintui 


aM'aa 


All  oM-^'t  Z\-~  ». 


niillius  iifotgafuictaim'iert  fcic'cia 

nijra  pfiSi;s:vt  116  ccterap  quoqj  ar 

tiu  abiumenta  befiberet.  Ili  injffi 

gianSis  Jnarmojcftatuis-alms  ej:/ 

abenb  c  molis  laboz  efl.-atia  fomta 

bf  imaginis  ratio  mceeiutoe  artifi' 

cis  manus  politi  operis  nito:  ejcpe> 

ctat.^tpiciitrc  maiubus  tabulc  o5 

milTcTabTonun.ccrerufhca  ob'fer/ 

uaiionebecCTptf  :coTomfuc»  mcrea 

toii'ifotcrtia  perquificilliinea  opero 

fis  elaboiata  tejrtrinis  -.miiTtiplieeiii 

maKnapjcttant.Tlonelbemquo/ 

que  bellop  vilitur  inftrumctisrDic 

fpKuia  fagittiseracuit-illi  valibus 

tbo:a]c  nigragemit  incube.^ltta'i/ 

us  XTUbivmbonis  tegmina,ppn)  la 

boeCotbi  infigenbaniercatur.tarn  .  , 

mulrf  arttbus ars yjn pglicit . 21(1  jafa  tiy/'n^nf 

noltri  labons  abfoUitio  logc  ab  fa'      .  ,/ 

cilioee'curriteucntuj.lCucmfolus  "^  nutuiof 

manuj  fupzeinoopcriiinpoiicG:in 

quo  nibil  be  becerncntiii  neeelTc  eft 

labo;arcc6feiifu.Quahbeteniboc 

iubiciii  multis  artibus  piobef  ejrcul 

til  vno  tame  cumulat  ejraminc.Hp 

periare  igitur  licet  quantu  nobis  in 

boc  ftubto  longis  tractua  oci;'s  la^ 

boz  a&ieccTit.2ln  rcrufubtilium  fit 

gas  exercitaic  mentis  velocitas  co^ 

pbenbat.vipieiunc  inacies  ozatio 

nis  ab  ea  que  fimt  ealigannbus  vn 

pebita  femenri/s  ejrpebienba  fuffici 

at.Oua  in  re  mibi  alieni  quoquc  iii 

bio;  lucra  qucruut.  JCuin  turtrarii 

quepCTiti(Timus  littera?:  pon'is  gr« 

icozationiscjrpertibus  quantusbc 

nobis  tubicare  aubeaut:  fo!a  taiitii 

pnunciationep:efcriberc..atn6  al^ 


-/  X^llMK* 


Fic 


First  page  ok  the   1488  roetihus 


26 

aXri^otti. 

RARA 

ARITHMETICA 

1 

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0ec^^avniW9. 


Xongitubo. 


Z^etragona. 


•ftatio  at(i5  cjcpolitio  bigeftf  fo:mu 

l^igitbuopmaUtey 
ra^poITt^  foznmlf  q 
Ifaciut  agulu:ab  vno 
a&.  lo.et.  lo.^ceben 
[tiarerpiciati'rbisfub 
jterio:e8ozt)fC8  copa 
rcnt.qni  fcilicet  a.4ansulum  incipi 
entes:invigeno6  terminu  ponunt: 
bu  pleic  it)  eft  pzima  fpccies  multipU 
CitdCia  often^imnita  vtputnua  p;i/ 


mil  fola  fiiperet  vnitate :  vt  5uo  \Miij 
fcc^^'  fcdm  binario  ^ll^Jua^at:v»tq/ 
tcrnari"  binariu .  teiti'^ter  tiu  tribus: 
vt  fenari'  tcrnariu.qrme  qrm  qtcr^ 
tiari)  nunierofitatetranrccn^at :  vt 
8  qtcrnariu  \z  p  ea&e  dicti  fcquenl 
Jcfeminoiis  pluralitate  pzetereant^ 
^Tvero  tati'angiiUis  afpiciatig  ab 
9.incboda  logitubine  la:itut)mc(^ 
triccnis  altritifeais  im!nerce]CtcMt: 
et  bic  cu  pnw  latitubiiie  et  logitubi/ 
necdparccururiplejcrpccied  multi^ 


Fig.  12.    Multiplication  table,   1488  boethius 


PRINTia)    BOOKS 


27 


venctijs     nuc  //  augufte    excellct    noniinalirfimus.//  Anno    diii 
.M.cccc.  Ixxxviij.  Men-//iis  maij  die  vigelima.'    {V.  48,  r.  Fig.  14.) 

Description.  4°,  15.1  X  20.8  cm.,  printed  in  double  columns, 
each  being  5.1  X  14.7  cm.  47  ff.  unnuml).  +  i  blank  =48  ff., 
40  11.     Augsburg,  1488. 

Editions.  Augsburg,  1488,  4°  (here  described)  ;  Cologne, 
1489;  Leipzig,  1490;  Venice,  'Opera,'  1491-92  (see  p.  28); 
Paris,  1496,  4°  ;  Venice,  1497  ;  ib.,  1499  (bearing  also  the  date 
1497),  fol. ;  s.  1.  a.  (the  Compendium  of  Muds,  _    ...  „,  ^^ 

c.  1500) ;  Paris,  1501,  fol.  ;  ib.,  1503  (seep.  29)  ;  /\/\^£k^ 
ib.,  1507,  4°;  ib.,  1 5 10  (see  p.  30),  fol.;  ib., 
151 1,  4°;  ib.,  15 14,  fol.;  Vienna,  15 15  (the 
Muris  '  Compendium,'  in  Tannstetter's  works)  ; 
Paris,  1521  (see  p.  31),  fol.;  ib.,  1522,  fol.; 
1528,  fol.;  Paris,  1530,  8°,  '  De  differentiis 
topicis  Hbri  quatuor '  ;  Basel,  1536,  8°;  Basel, 
1546,  'Opera,'  fol.  ;  ib.  and  Paris,  1549;  Basel,    F'«-  13-    F'^ur- 

,     .       ^.  T  T      c    u       I     r  'CO  ATE    NUMBERS, 

1553,  'ajectis  exphc.  per  J.  Scheubelium,  8  ;  ,^§8  boethils 
Paris,  1553;  Basel,  1570;  Venice,  1570,  'Opera,' 
fol.  There  are  undoubtedly  various  other  editions  of  the  Arith- 
metic, or  the  Epitome  by  Faber  Stapulensis,  these  works  often 
being  bound  with  such  treatises  as  the  Arithmetic  of  Jordanus 
Nemorarius  Murhard  (I,  160)  and  Rogg  (p.  137)  mention  an 
edition  of  Faber's  'Compendium,'  s.  1.,  1480,  but  it  is  not  given 
by  other  bibliographers.  (See  Hain,  I,  468  ;  Brunet,  J\Ian.,  I, 
1059;  Graesse,  Tresor,  I,  464;  Riccardi,  I,  i,  159;  Boncom- 
pagni,  Bullctino,  XII,  148.) 

The  text  is  practically  that  followed  by  Friedlein  in  his 
standard  edition  of  the  'Opera'  of  Boethius  (Leipzig,  1867). 
except  as  to  numerals.  Here,  as  in  the  later  manuscripts,  the 
Arabic  characters  have  replaced  the  Roman  of  the  original 
text.     (See  Fig.  12.) 

The  arithmetic  of  Boethius  was  based  upon  the  Greek  work  of 
Nicomachus  (fl.  c.  100  a.d.),  and  related  only  to  the  theory  of  numbers, 
the  'ApiOfjirfTiKi),  as   distinguishetl   from  the  ])ractical  calculations,  the 


28 


RARA   ARITHMETICA 


Aoyto-TtKT/,  and  from  the  later  algorismus  (p.  5).  Boethius  gave  an  elabo- 
rate theory  of  ratios  and  devoted  much  attention  to  figurate  numbers,  such 
as  the  triangular,  square,  pentagonal,  and  cubic.  (See  Fig.  1 3.)  The  work 
was  the  standard  in  the  Church  schools  throughout  the  Middle  Ages. 


•)i!opomonalaaogeomcm<fl. 


T^iopoiiionaluae  anoonica. 


8  fupcraf  .Qiiatuo! 
fniqbm  ccionariue  a  &iioi»cnano 
rinnf :euoOcnari)  Krcia  pe  c .  £  i  li 


p«rtoniiriBmc6iunimpUcee.  144- 
fut.Qftfifcwtrcmirorcs  mltipUcct: 
fMfcjct-  ii.faaci'iquonumCTO 
144-BupUiecft.  "^nacnic 


muli>ifl:efTCTon  ^fonantia,  Bej:  io 
«&.9.«|.8.a».  11  coporari  re^^ut 
klciaalxaippowoncileb  ^rapcncc 
r7mpbonia.^^uo6edm^)oa^  fcna/ 
ri^^S^erari6llpliaJponionc:  ftS 
6f  apafon  rrmpbonia  camint.Doo 
*o  t.^ipK  oira  (e  nKbii  conRScraa 
cp«^oumn«W•gmmulico  bio£)u 
lamine  lonoe  vocal  .qiij  oinj  mua/ 
comronoiumcnfiira  coinunis  eft, 
Omnui  cnl  eft  fonue  iflt  parmffim' 
Un6e  noni  eft  q6  ftiaecffcron  zb\» 
p^u  oibnantiarii  [onos  biifcretia 
cft:ncul  inter  fctquiicrcia  z  fcfqual^ 
ceiajtpouiDiufolacepocdous  &if 
'I  ^efaipt^oia  fob/ 


5erqiiainTf4po;non<:s. 


cnrcmoii  meWoycB  mttiptoltioea 


f  init  ariibmcricaBocri)  bcncr^ 
ulfa  ac  toeli  (bibio  emenSata  3Jm 
matOiifcB  fmino)!.  pirtTa  per  f  rbar^ura[6oltv^ri  fo- 
leTtifTimiepmifl  t^uftnaT  mira  im' 
pamebiane:quanu^venenje  nuc 
augiift^  erceUa  nominatilfunua. 
annoOra.fD.cca.lmriij.iOciv 
ri8;naij  &ievtgefima. 


a  b  c  a  c  I      omiicsquaiani. 


£V~^ 


Scfgtenia  Sefgtcrda 

£pocSoita 
fciaiefUTOn  Isiaicirero 


Fig.  14.    Last  two  pages  of  the    1488  boethius 


BOETHIUS. 

Ed.  pr.  of  the  Arithmetic,  1488.  Venice,  1491-92. 

See  p.  25. 

7>VA'.    See  Fig.  15. 

Colophon.  '  Impreffis  venetijs  per  Joanne  de  Forli-//uio  et 
Gregorium  fratres.  Anno  falutis  .M//(;ccc.lxxxxj.  die  xxvj,  men- 
fis  Martij.'     (F.  352,  r.) 

On  f.  256  (220  as  numbered  in  the  book),  at  the  end  of  the 
Geometry  is  the  following :  '  Venetijs  Impreffum  Boetij  opus  p 
Joane5  2  Gre//goriii  de  gregorijs  fratres  felici  exitu  ad  fine  vfq5 
pductu //accuratiffimeq5  emedatu  Anno  humane  reftaurationis. 
//  1492.  die  .18.  Augufti.  Auguftino  Barbadico  Sereniffi//mo 
Venetiarum  principe  Rem  pu.  tenete.' 

Description.  FoL,  21.7  x  32  cm.,  printed  in  double  columns, 
each  being  6.6  x  24.  i  cm.  3  ff.  unnumb.  +  345  numb.  +  i  blank 
=  349  ff.,  66-70  11.    Venice,  1491-92  (see  the  colophons). 


PRINTr:i)    HOOKS  29 

Editions.    This  is  the  i-ditio  princcps  oi  the  works  of  Hoclhius. 
For  other  editions  see  p.  27. 

10{c  runt  opera  »octil:qu{iit  boc,voIuminc  cotitfticntur. 

7fntx>rpbyritgrj«oscnaCJicto2inotTannatam  ^dinoprtma. 

llJ^nJljrniillWna  JOoctioiprotra.ifUtani  cduio  feconda 

^n  cafbcgonaa  arlftotclia  {ditto  vna. 

inlllbJumanftottlie-ocintcrprttanoiiccdmoprtma. 

fn  ?ur.dcm  iib:uni  t>cintcrprftanoiic{dmorccunda. 

f^c  f  iuilloiilbus  libCT  vnue. 

©c  w:finitionibuolibcr  vnua. 

adcatbcsoncoerrlioprmoomtroducno 

xCotmnciitariojuminXopicaCiccroiiiaUWlfcj:. 

©c  tmcrciitiio  ifoplcialibn  iiuatnjor. 

#Dc  frllofiifiiio  catlxsorico  libn  cuo. 

^)c  Mtof;iriiio  bipoibctico  librt  tiuo> 

©ctnnitafclibripuo 

©cbcbdoniadibiit'lllbcrvnufl: 

©evniiatc  1  viiolibcr  vniw. 

Conn-a  cutbicbcii;!  TRcftortum  Dc  tmboa  naturto:  1  vna  pcrfotu 

(brtftilibcivima. 
Z>e3ritbnicrtcaad  Ipatritiam  (tmmacbnmblnl  coo- 
ScilliurialibnqaintB. 
,^c  -©comctrialibn  odo. 
25c  pbiiofopbif  confolanonc  llbrl  quintp. 
ipcrcboiariumt»irclpUiM  liber  vnue. 


491-92     BOETHIUS 


BOETHIUS.  Ed.  pr.  of  the  Arithmetic,  1488.  Paris,  1503. 
See  p.  25. 
Title.  'In  hoc  Hbro  contenta.// Epitome/ compcndiofaq.^  in- 
troductio  in  libros // Arithmeticos  diui  Seuerini  Boetij:  adiecto 
fa-//miUari  commentario  dilucidata.// Praxis  numerandi  certis 
quibufdam     regulis //conftricta.// Introductio    in    Geometriam 


30  KARA   ARITHMETICA 

breuiufculis  an-//notationibus  explanata.  fex  libris  diftincta.// 
Primus  de  magnitudinibus  et  earu  circuftan-//tiis.//  Sc'dus  de 
cofequetibus/  cotiguis/  &  c6tinuis.//Tertius  de  punctis.//Quar- 
tus  de  lineis.//Quintus  de  fuperficiebus.//Sextus  de  corporibus. 
//  Liber  de  quadratura  circuli.//  Liber  de  cubicatione  fphere. 
// Perfpectiua  introductio.// Infuper  Aftronomicon.'    (F.  i,  r.) 

Colophon.  '  Abfolutum  in  almo  Parhifiorum  ftudio/  //Anno 
diii  qui  numero  definuit  omnia//  1503.'     {F.  48,  r.) 

On  f.  84,  V.  is  the  following:  '  Geometrici  introductorij:// 
fexti /etvltimi  libri  finis.// Editi  anno  domini //millefimo  quin- 
//getefimo  pri//mo  :  vicefi-//ma  quin//ta  no-//uem//bris.'  And 
on  f.  Ill,  v.,  the  following:  '  Id  opus  imprefferut  Volphgangus 
//  hopilius  et  Henricus  ftephanus  //  ea  in  arte  focii  in  Almo 
pari-//fiorum  ftudio  Anno  Chri//fti  Celorum  totiufqj // nature 
coditoris.// 1503.    Die   vice//nmafepti-//ma  Iu-//nij.' 

Description.  8°,  20x26.8  cm.,  the  text  being  15.6X  22.3 
cm.  48  ff.  numb,  in  the  arithmetical  part,  113  in  all  (i  blank), 
54  11.     Paris,  1503. 

The  editions  of  Boethius  differ  more  or  less  in  the  combinations  of 
works  which  they  contain.  See  p.  27.  This  is  the  first  edition  of  the 
Jacobus  Faber  Stapulensis  and  Jodocus  CHchtoveus  '  Epitome.' 

BOETHIUS.     Ed.  pr.  of  the  Arithmetic,  1488.       Paris,  15  10. 

See  p.  25. 

Title.  This  is  substantially  identical  with  that  of  the  1503 
edition  already  described. 

Colophon.  '  CAbfolutum  in  almo  Parifiorum  ftudio/  //  Anno 
domini  qui  numero  definiuit //omnia  1503.  Et  emiffum  ex  offi- 
//cina  Henrici  ftephani  Anno  //  Christi  faluatoris  //  omnium 
1 5  10  de-//cima  quinta//die  Mar-/Aij.'     (F.  xlviii,  r.) 

Description.  Fol.,  19.5  X  27.9  cm.,  the  text  being  15.4  X  25.6 
cm.     48  ff.  numb.,  46-56  11.     Paris,  15  10. 

This  is  one  of  the  editions  containing  the  commentary  of  Jodocus 
CHchtoveus  on  the  'Epitome'  of  Boethius  by  Jacobus  Faber  Stapulensis. 
(Seep.  27.) 


PRINTED    BOOKS  31 

BOETIIIUS.     Ed.  pr.  of  the  ArithmcUc,   1488.       Paris,  1521. 

See  p.  25. 

Title.  '  Divi  Scvcri-//ni  l^oetii  Arilhmctica,//  dvobvs  discrcta 
libris  ;  adie-/A"to  commentario,  mysticam  nvme-//r')runi  a|)])lica- 
tionem  pcrftrini^cnle,  declarata.// 
(Woodcut  with  initials  of  the 
printer:  S.D.C.).  Voenundatur 
apud  Simonem  Coli-Z/nsevm,  e 
regione  fchoke  Decretorum.' 
(F.  i,r.) 

Colophon.    '  ExcLidebat  Simon 

^  ,.  ^    .,.     .         MDXXI 

Cohnseus,  ranliis,  Anno 

1521 

Quinto//Idus  lulias.'  (F.  139,  \ .) 

Description.  FoL,  19.9  x  28.2 
cm.,  the  text  being  13.9x26 
cm.  4  ff.  unnumb.  +  136  numb. 
=  140  ff.,  40-52  11.    Paris,  1 52 1. 

This  is  the  first  edition  of  Boethius 
with  the  commentary  of  Girardus 
Ruffus  :  '  CCiirardi  Rvfifi,  in  duos 
arithmeticse  Boetii  libros,  commen- 
tarivs.'  (F,  5,r.)  This  commentary 
greatly  exceeds  the  text  in  extent, 

and  as  to  ponderosity  it  leaves  little  to  be  desired.    As  a  piece  of 
typography,  however,  this  is  one  of  the  best  editions  of  Boethius. 

ANIANUS   AND  JOHANNES   SACROBOSCO. 

Ed.  pr.  1488.  Strasburg,  1488. 

Anianus  was  a  fifteenth-century  astronomer  and  poet,  of  Strasburg. 

Johannes  de  Sacrobosco  (Sacrobusto,  Sacrobuschus,  Hoi.vwood, 
HoLYBUSH,  HoLYWALDE,  HoLYFAX,  HALIFAX)  was  bom  at  IIalifa.\  (Holy, 
wood),  Yorkshire  ;  died  at  Paris  in  1244  or  1256.  He  studied  at  O.vford  and 
lectured  at  Paris.     He  wrote  on  astronomy  and  algorism. 

Title.  '  Copotus  manua//lis  magri  aniani. // mctricus  cfi 
;ometo//Et  algorifmus.'  (F.  i,  r.  Fig.  17.)  F.  45  begins, 
'  Incipit  textus  algorifmi.' 


32  RARA   ARITHMETICA 

Colophon.  '  Impreffum  Argn.  per  Johamem  pryli.// Anno  do- 
mini  .i488  .is.  kail.'  decembris.'     (F.  44,  r.) 

Description.  4°,  13.9  X  19.5  cm.,  the  text  being  8.8  x  14.4 
cm.  53  ff.  numb.  +  2  imnumb.  =  55  ff.,  31-34  11.  Strasburg, 
1488. 

Editions.  The  editions  of  Sacrobosco's  Algorisvins  were  as 
follows  :  Strasburg,  1488,  4°  (here  described)  ;  s.  1.  a.  (Venice  .\ 
1490  .'),  4°  ;  Venice,  1501  (see  p.  35);  Paris  (edited  by  Clich- 
toveus),  1498;    ib.,  1503;    ib.,   15 10;    Vienna,  15 17;    Cracow, 

CXopomsmamta 
€ta!0mifittas 

Fig.  17.    Title,  first  edition  of  .anianus 

1504,  1509,  1521,  and  1522;  Paris,  1522;  Venice,  1523  (p. 
35);  Antwerp,  1582.  The  editions  of  Anianus  were  as  follows  : 
Strasburg,  1488,  4°  (here  described) ;  Lyons,  1489  ;  ib.,  1490, 
4°  ;  ib.,  1 49 1,  4°  ;  ib.,  1492  (two  editions),  4°  ;  Rome,  1493,  4°  ; 
Paris,  1494,  4°;  ib.,  1498,  4°;  s.  1.  a.  (Paris.?,  c.  1495,  4°,  see 
p.  33);  s.  1.  a.  (Basel,  c.  1500),  4°;  Rouen,  s.  a.  (1502),  8°; 
Paris,  1 501,  4°;  ib.,  1502,  8°;  Lyons,  1504  (see  p.  35);  ib., 
1509,  4°;  Paris,  1508;  ib.,  151 1;  ib.,  15 15;  ib.,  15 19,  4°; 
ib.,  1529;  ib.,  1530,4°;  Lyons,  1540,4°;  p-rankfort,  1549; 
Wittenberg,  1550;  ib.,  1568;  Antwerp,  s.  a.  (c.  1558);  ib., 
1559.     There    are    probably   others    s.   1.  a.,    and    some    appear 


PRINTED    HOOKS  33 

under  the  name  of  Sacrobosco.  Of  this  first  edition  Honcom- 
pagni  knew  only  the  Munich  C(^py.     {/htilctino,  XII,  126  n.) 

The  first  part  of  this  rare  book  is  the  Coinpoiiis  Manitalis  of  Anianus, 
and  the  second  is  the  A/goris/ni/s  of  Sacrobosco,  described  later.  It  is 
probably  the  first  book  on  mathematics  printed  in  Strasburg  (but  see 
p.  10),  and  it  is  the  first  edition  of  each  of  the  two  treatises  mentioned, 
and  the  first  printed  work  on  the  computus,  the  arithmetic  of  the 
Church  calendar.  In  the  work  of  Anianus  appears  for  the  first  time 
in  print  the  original  of  the  rhyme  beginning 

'  Thirty  days  hath  SeptLMiiber,' 

the  English  version  of  which  is  said  to  have  been  first  published  in  the 
1590  edition  of  Grafton's  Chronicles.  It  also  appeared  in  an  arith- 
metic published  anonymously  in  1596.  The  Latin  form  of  Anianus  is 
as  follows  : 

'Junius  aprils  feptember  et  ipfe  nouember 
Dant  triginta  dies  rebcjuis  fuuadditur  vnus, 
De  quorum  numero  februarius  excipiatur."    (F.  B  8.) 

It  is  found  in  various  forms  in  other  computi,  manuscript  and  printed. 
Anianus  also  gives  for  the  first  time  in  print  the  astronomical  formula : 
*  Sunt  Aries,  Taurus,  Gemini,  Cancer,  Leo,  Virgo,  Libraque,  Scorpio, 
Arcitenens,  Caper,  Amphora,  Pices,'  which  appeared  in  the  works  of 
Bede  under  the  title  '  Verfus  Prifciani,  de  Astronomia,'  as  '  Hinc  Aries, 
Taurus,  Gemini,  Cancer,  Leo,  Virgo,  Libra,  Scorpius,  Arcitenens,  Capri- 
cornus,  &  Vrna,  Qui  tenet  &  Pifcis.'    (1563  edn.,  vol,  I,  517.) 

ANIANUS.     Ed.  pr.  1488.  S.  1.  a.  (Paris  .?,  c.  1495). 

See  p.  31. 

Title.  '  Copotus  cum //commento.'  'Liber  cpii  Compotus 
infcribitur  :  vna  cum  figuris  et  ma/  //nibus  neceffariis  tam  in 
fuis  locis  qj  in  fine  libri  pofitis.// Incipit  feliciter.'     (F.   i,  r.) 

Description.  4°,  13.1  X  19.4  cm.,  the  text  being  9.3  X  13.1cm. 
39  IT.  unnumb.  +  i  blank  =  40  ff.,  35  11.    S.  1.  a.  (Paris  .?,c.  I495)- 

Editions.  See  p.  32.  This  rare  and  interesting  edition  was 
published  about  1495,  possibly  by  Mich,  le  Noir  at  Paris, 
although  there  is  no  date  or  place  of  publication  given.  It  is 
one  of  the  best  examples  of  the  mediceval  computus  that 
appeared    in   print.     Unlike   the    first   edition    (jx    31,    14SS)    it 


34  RARA   ARITHMETICA 

contains  a  number  of  illustrations  showing  the  use  of  the  hand 
and  fingers  in  assisting  in  calendar  reckoning,  the  title  of  '  Com- 

Zfberoof  £ompoms  {nfcribtajrrvnaciimfistiris  etma/ 
mbiis  nccclfem^  tarn  in  (m»  lods  ^  in  fine  Ubii  pofiito. 
5iiapftfeUotcr  - 

^jcoztaeftiuftollbnjlmmajftaver   r^^^  ..y 

ba  poffcnt  OiipUdtcr  confideran .  p  nmo  pnt  -" '  '^^ 
Old  oe  oeo  q  dl  lujc  va^.  loeo  Dicebat  oauid*  / 

latotiacjliuJlo.ietoeillaluce  oiat  5<^b3.(. 
j£rat  lu]c  vera  ^  iTlumiat  omne  borain^  vcnie/ 
te  frtbuitc  mimdii.  Sccudo  pomi  oe  fdentia, 
Ct  tfidi  Itlt  (liWiT  focttte  rcddete  luddi.qwa  fadt  bomme  fde/ 
tern  eflc  loddu.'jii  qUibus  verbis  ad  comcndanone  fcie  ouo  bjc 
ufter  tangiimr.primo  e;n  tangif  fd^tiealritudo  pzcaofa  p  boc 
quodoidtlujc.  Scdjndo  largimdo  gloziofa  per  boc  cp  oidt  oi/ 
ra  ell.lSnmu  pzobarar  audoiiratc  et  ratioe.aiKtoncate  5ftdoi< 
nc oicmris/^aSSae;! fens indcfidens Jjoti'tane via.fai fal/ 5 c  ^ e?^ fT^a>' 
uaton0co^tfo.Ba:{5enc.illudc,^vaUdi  ap:edofamquod  MY--/;:''^'^ 
©c  inoalido  i  imperftcto  fadt  validum  i  pfecrutn.foctia  e  boiuf      (/ ' 
modi.ergo  zc.  imioz  dt  mamitlta.  minoz  occlaramr  p  pbin  rer/ 
cfo  oe  aia  He  oiccnte:  Hia  in  pndpio  foe  creatt'onis  e  taiv^  tabu 
la  rafa  in  qua  niba  oepicta  elt.oepmsibilie  m  fderiis  z  virturt/ 
bus.*l>zinwm2?pb3|aaao2it3tc  boeniiratoe in pzologoariC    ^_ 
mcmcefec?enSa"e  coy.  que  vera  funt  z  t'mcmutabilio  eiTeim'e  no  2- 
riraq5  compbenfione  vcritaris.Barioe  ficnllud  eH  tanop  (md 
bonii quod babet largirion^ glonofam fcia e bmoi . ergozc.ma,    ^ ^ 
<02 dl vera.mtnoz ^bamr poilfinitione fcicne q talis c.Sciadt  3  ' 
(Jda  babitus  aic  ronalie  no  innatus  fed  accifims  oim  buanair  re 
rnmKndagamjc  z  totius  bumane  vite  gubcmam']c/cv'  r<ia  fit  ba/ 
bitos  pt5.qz  fcia  ell  aliquod  ejcillens  in  aia.fed  omne  illud  quod 
cit  in  ala  aut  c  bitiis  aut  potetia  aut  palTio.  €tbcc  teilat  arilto. 
tn  fcciuJo  eibicott.q'  fcia  no  fit  paTio  pt5  qz  palTiones  funt  in  vo 
luniate  fcia  no  c  bmoi.ergo  zc  .cp  no  fit  potetia  pt5.q:  ql;b5  po/ 
ten  -ia fit  a  natura.  flcit  irafcibiliG.x  cocjpifabilis.t  (ic  rdinqui 
wr  ^  no  fit  potentia.  ($  fit  bltus  ale  ronalis  pt5 p  pdiaa-cp  aiiJ 

a  a 

Fig.  18.    Beginning  of  anianus,  edition  ob^  c.  1495 

putus  manualis  '  being  thus  justified.  This  differs  from  the  1488 
edition  in  the  notes  of  the  commentator.  The  original  text  is, 
however,  substantially  unchanged. 


PRINTED    BOOKS  35 

ANIANUS.     Ed.  pr.  1488.  Lyons,  1504. 

Seep.  31. 

Title.    •Compotus  cu //commento.'     (F.  i,  r.) 

Coloplion.  '  CLibcr  compoti  cum  comcnto  finit  fclicitcr  //  Im- 
preffus  Lugduni  per  Claudiu  nour//ri.  Anno  domini  .M.ccccc.iiij. 
//die  .iiij.  Octobris.'     (F.  32,  r.) 

Description.  8°,  14.5  X  20  cm.,  the  text  being  8.9  x  14.  i  cm. 
32  ff.  unnumb.,  34-40  11.     Lyons,  1504. 

See  p.  32.  In  this  edition  there  is  a  curious  misprint  in  the  calendar 
verse  given  on  p.  25  (f.  13,  r.).  It  here  begins  'Julius  (instead  of 
Junius)  aprilis  feptembre  &  ipfe  nouember.'  Like  the  edition  of  c.  1495 
"(p.  33),  this  is  well  illustrated.  It  differs  from  the  1488  edition  in  the 
notes  of  the  commentator,  but  is  practically  identical  with  that  of 
c.  1495. 

JOHANNES   SACROBOSCO.    Ed.  pr.  1488.    Venice,  150 1. 

See  p.  31. 

Title.  '  Algorifmus  Domini  Ioa-//nis  De  Sacro  Bufco  // 
Nouiter  Impreffuj.// Cum  Gratia  Et  Priuilegic'     (F.  i,  r.) 

Colophon.  '  Impreffum  Venetijs  per  Bernardinum  Venetum  // 
De  Vitalibus:  Anno  Dni  .M.CCCCC.L// Die  Tertio  Mefi. 
Februarij.'     (F.  8,  r.) 

Description.  4°,  14.  i  X  19.6  cm.,  the  text  being  10.5  X  16.8 
cm.     8  ff.  unnumb.,  39  11.     Venice,  1501. 

See  p.  32. 

JOHANNES   SACROBOSCO. 

Ed.  pr.  1488.  S.  1.  a.  (Venice,  1523). 

See  p.  31. 

Title.  'Algorifmus  Domini  Joannis//de  Sacro  Bufco  noui- 
//ter  impreffum.'     (F.  i,  r.) 

Colophon.  '  Impreffum  Venetiis  per  Melchiorem  Seffam  & 
Petrum//de  Rauanis  Socios.  Anno  domini  .M.D.XXIII.// 
die  .XXIIII.  Octobris.'     (F.  8,  r.) 

Description.  4°,  14.8  X  18.6  cm.,  the  text  being  10.3  X  16.8 
cm.     8  ff.  unnumb.,  40  11.    S.  1.  a.  (Venice,  1523). 


36  KARA   ARITHMETICA 

JOHANN    WIDMAN  (?).     Ed.  pr.  c.  1488.     Leipzig,  c.  1488. 

Bom  at  Eger,  Bohemia,  c.  1460.  He  was  a  student  at  Leipzig  in  1480, 
A.B.  in  1482,  bachelor  of  medicine  in  1485,  A.M.  in  i486.  He  evidently 
received  the  doctor's  degree  about  the  same  time,  for  he  wrote  a  medical 
work  in  1497  with  the  title, '  Tractatus  clarissimi  medicina  ^  doctoris  Johanis 
widman  . .  .  de  pustulis  .  .  . '  That  he  gave  lectures  on  algebra,  possibly  the 
first  at  Leipzig,  is  proved  by  a  passage  found  by  Wappler  in  an  old  Dresden 
manuscript :  '  Quare  hodie  hora  secunda  post  sermonem  atque  Baccelaureo- 
rum  celebrata  disputatione  Magister  Jo.  W.  De.  Eg.  Aporismata  at  Regulas 
Algobre  resumpturus  pro  hora  atque  loco  conuenienti  cum  audeturis  con- 
cordabit . .  .' 

Title.  '  Algorithmus  Linealis.'  (F.  i,  r.)  'Ad  euitadum  mul- 
tipli//ces  Mercatorum  erro-//res  et  alteri^ .  .  .'  (F.  2,  r.) 

Description.  4°,  14  X  19.5  cm.,  the  text  being  7.8  X  14.5  cm. 
I4ff.  unmimb.,  31-34II.   Initials  in  red,  by  hand.    Leipzig,c.  1488. 

Editions.  Leipzig,  s.  a.  (c.  1488,  here  described);  ib.,  possi- 
bly 1490,  1493;    1516;    15  17,  and  several  others. 

This  rare  treatise,  the  first  printed  work  on  calculation  by  the  aid 
of  counters  ('apud  noftras  appellata  eft  calculatio '),  is  of  unknown 
authorship,  but  was  probably  written  by  Widman.  {Abhafidlungen,  V, 
152.)  At  the  end  of  the  book  is  the  device  of  Martin  of  Wurzburg 
(Martinus  Herbipolis),  and  the  book  was  printed  by  him,  probably  c. 
1488.  After  a  brief  introduction  on  the  use  of  counters  ('projectiles' 
as  they  are  here  called),  the  author  treats  of  the  following  topics  :  De 
Additione,  De  Subtractione,  De  Duplatione,  De  Mediatione,  De  Multi- 
plicatione,  De  Diuifione,  De  Progreffione,  De  radicum  extractione,  De 
radicum  extractione  in  Cubicis.  The  book  closes  with  the  words :  '  Et 
tantij  de  Radicum  extractione  et  vltima  huius  Algorithmi  fpecie  Et  p 
confequens  de  toto  Algorithmo.'  There  are  no  applied  problems,  and 
the  only  computations  with  abstract  numbers  are  performed  'on  the 
line'  (i.  e.  by  the  'projectiles'  on  a  line  abacus),  whence  the  name 
'  Algorithmus  linealis.'    The  work  is  illustrated  by  woodcuts. 

Other  works  0/1488.   Suiseth  (p.  10,  c.  1480)  ;  Borghi  (p.  16,  1484). 

JOHANN    WIDMAN.     Ed.  pr.   1489.  Pforzheim,  1500. 

See  above. 

Title.  '  Behennd  vnd  hupfch  //  Rechnung  vff  alien  //  kauff- 
manfchafften.'     (Woodcut  of  a  schoolroom.)     (F.  i,  r.) 

Colophon.  '  Gedruckt  zu  Pfortzheim  von  Thoman  //anfzhelm 
Im  Jubcl  Jar  als  man  zalt  1500 //Got  fey  lob.'     (F.   163,  v.) 


PRINTED    1K)()KS 


37 


Description.  i6°,  lO  x  13.2  cm.,  the  text  beinj]^  6.5  X  10.4  cm. 
II  ff.  blank  +  i  unnumb.  +  162  numb.  =  174  ff.,  26  11.  Pforz- 
heim, 1500. 

Editions.  Leipzig,  1489,  8°;  Pforzheim,  1500,  8°  (here 
described);  ib.,    1508,  8°  (p.  39);   Hagenau,    15  19,  8°  (p.  40); 

tl 

^leif^  ale  vil  in  fic^  bcf^Iieffen  old  bie  wffcl 
imqu*brAt/Trclcf?e  bannbie  anbcr  ^efc^t  ijl 
Ab  bArifi  l?ie  ^cr n Ad>  ietlt4?e  an  ix  felb(l  foitn 
f  terltc^cn  befc^reibert  i(t . 


16liZ|i8^4j5ol5l7 


'4p'|z8|55l42|45l8 
i6fz4|5z|4o|4a|56|^9_ 


I9l^jz7i56|4^l54l65|7r|9i|  _ 

£errttpolmu:flei0  baa  em  ma  I  em  @o  twit 
^ir  alle  rec^nung  0emetn 

f[^|5  ^f75  |6>  18  |9  I 
^14  16  18  iio|iZli4iiig;i-8) 
lXirMl'5l'82i|^4fe7| 
-rt3Ti|rgg'clz4|z8|5i|f6i 


|33iJ|zo^55g5^ 
|6lTili8blJ^^42W4i 
|7Mzi;z8|55!4-|4  ^5  ^6^1 
|3^^544o}4j26ig72| 
7ni8'2l^lB5K4'6^f7z§i) 

Fig.  19.    From  the   1500  \vidm..\n 


Augsburg,  1526,  8°  (p.  40).  The  title  of  the  first  edition  was 
as  follows  :  '  Behede  vnd  hubsche  //  Rechnung  auff  alien  // 
kauffmanschafft,'  and  the  colophon   (f.  236,  r.),  '  Gcdruckt  In 


38  RARA   ARITHMETICA 

der  Furstlichen  Stath  //  Leipezick  durch  ConradQ  Kacheloffen 
//Im  1489  lare.' 

^u  m A<^  yerfic^d  fiir  (ic^  felb  mit  femcm  gelt 
rt<i4^berre0elfohimt(ib^ien<xc^  flat 

pzoflotet      305      .5        'reum^ff 
jToret        ff  Z04  f    5  f^ft  9      1 161  pi 


2 


primer  A        65         10        7^ 

©ecAnba       40       7 
SXfc<i)[d 


20.    Exchange,    from  the  1500  widman 


This  is  the  second  edition,  and  is  even  more  rare  than  the 
first.  It  was  unknown  to  Boncompagni  when  he  printed  his 
'  Intorno  ad  un  Trattato  d'Aritmetica  di  Giovanni  Widmann  di 
Eger'  in  the  Bidletino,  IX,  188,  the  best  discussion  of  this 
arithmetic  that  has  appeared. 


PRINTED    BOOKS  39 

Widman's  arithmetic  was  the  first  great  German  textbook  on  the 
subject,  although  minor  works  had  already  appeared  before  1489.  It 
is  in  the  main  a  practical  treatise,  with  good  problems,  and  it  set  the 
standard  for  Germany  much  as  Borghi's  book  did  for  Italy,  Among  its 
noteworthy  features  is  the  use  of  the  plus  and  minus  signs  for  the  first 
time  in  a  printed  work.  (See  Fig.  21.)  These  are  not  used,  however, 
as  signs  of  operation,  but  as  symbols  of  excess  or  deficiency  in  ware- 
house measures.  The  book  is  illustrated  (see  Fig.  20)  with  pictures 
showing  mercantile  customs,  and  with  geometric  diagrams.  Widman 
acknowledges  his  indebtedness  to  men  like  Sacrobosco,  '  als  da  lert 
Joannes  defacrobufto  vii  ander  mer,'  although  his  work  shows  no  depend- 
ence upon  the  '  Algorismus  '  named. 

After  the  '  Inhalt  difz  buchs  in  einer  gemein,'  Widman  devotes  2^ 
pages  to  '  Numeratio,'  2^  to  'Additio,'  2^  to  '  Subtrahiren  '  (including 
denominate  numbers  in  these  topics),  i  to  '  Dupliren,'  2  to  '  Medieren ' 
(i.e.  multiplying  and  dividing  by  2),  ii^  to  '  Multipliciren,'  5  to  divi- 
sion, 2i  to  progressions,  and  14  to  roots  (4  referring  to  cube  root).  He 
then  takes  up  fractions  in  the  same  order,  this  work  being  followed  by 
compound  numbers  and  proportion.  He  then  gives  a  large  number 
of  type  problems,  regiilae  as  he  calls  them,  although  they  are  not  stated 
in  the  form  of  rules  as  we  now  know  them.  These  include  the  '  Regula 
detri'  (rule  of  three,  treated  as  distinct  from  proportion),  and  the  regulae 
fusti,  detri  conversa,  positionis,  equalitatis,  legis,  augmenti,  plurima, 
sentencianmi,  suppositionis,  residui,  excessus,  collectionis,  quadrata, 
cubica,  reciprocationis,  lucri,  pagamenti,  and  alligationis. 

Other  works  of  I48g.    Boethius,  p.  27,  1488  ;  Anianus,  p.  32,  1488. 

JOHANN  WIDMAN.     Ed.  pr.  1489.  Pforzheim,  1508. 

See  p.  36. 

Title.  'Behend  vnd  hupfch // Rechnung  vff  alien // Kauff- 
manfchafften.'     (F.  i,  r.) 

Colophon.  'CGedruck  zu  Pfhorzheim  von  Thoman//Anfzhelm 
Im  iar  als  man  zalt  1508.'     (F.  161.) 

Description.  8°,  9.7  X  13  cm.,  the  text  being  6.5  X  10  cm. 
7  ff.  blank  +161  numb.  =  168  ff.,  26  11.     Pforzheim,  1508. 

Editions.  See  p.  37.  This  is  the  third  edition  of  this  famous 
arithmetic,  and  is  by  the  same  publisher  as  the  second  (1500), 
but  is  from  different  type.  It  is  about  as  rare  as  the  first 
edition. 


40 


RARA   ARITHMETICA 


JOHANN    VVIDMAN      Ed.  pr.  1489.  Hagenau,  15 19. 

See  p.  36. 

Title.  '  Behend  vnd  hiipfch  //  Rechnung  vff  alien  //  Kauff- 
manfchafften.'     (F.   i,  r.) 

Colophon.  'Getruckt  zu  Hagenaw  durch  Thoman // Anshelm. 
Im  iar  als  man  zalt  1 1  \S^9-      (F.  i  5  i ,  r.) 

Description.  8°,  9.7  X  14.3  cm.,  the  text  being  6.8  x  11  cm. 
I  f.  unnumb.  +  151  numb.  =  152  ff.,  20-29  11.     Hagenau,  15  19. 

Editions.  See  p.  37.  This  is  the  fourth  edition  of  Widman's 
Arithmetic.  As  an  inscription  on  the  fly  leaf  says,  this  copy 
was  presented  to  Prince  Baldasarre  Boncompagni  by  Ludwig 
Kunze  as  a  '  liber  rarissimus.' 

See  p.  39. 


JOHANN   WIDMAN. 

See  p.  36. 


Ed.  pr.  1489. 


Augsburg,  1526. 


Title.  '  Behennde  vnnd  //  hiibfche 
Rechnug  auff  alien  //  Kauffmanfchaff- 
ten.'  (Woodcut  showing  two  men 
seated  at  a  reckoning  table.)  '  M.  D. 
XXVI.'     (F.  I,  r.) 

Colophon.  '  Getruckt  zu  Augfpurg 
durch // Haynrich  Stayner// M,  D, 
XXVI.'     (F.  192,  r.) 

Description.  9.4  X  13.7  cm.,  the  text 
being  G.y  y.  11  cm.    2  ff.  unnumb.  +  190 

trrtt)Kr»o.HTJ9.Vt.0  61<:vbert4'Ti      numb.  =  I92  ff.,  25  11.     Augsburg,  I  526. 
tt.nimfpj.d)iooH,&a6.|Iem5entrtec  c^.VV....      C„^  ,.     .»       Jhig  COpy  haS 


4  +  7    W!tb»tisot»vf< 

3   -j-  30    d)ei?/Qorutruct 

4 1  i)    lMc2crictiier»nt> 

3-1-44  ttwnt)  waeaug 
3   -f-  2z  — i(l/0«et(l  rtit> 

3 1 1  tfc  nu6&jf«G6c(crt< 

3  +  S*    fcctonn&ttjct&en 

4 16    4?3  9K,  CQ« 

3    4-  44  6u  bi'c  2cnl>tner 

3    4-    *9  56  K>  Seni<»*«e 

3  — T—i  z  ^4(i»)^^rt^t)<^a  / 

3    4-    p     -}-  feae  i(i  twee 

fe<jrjii2J5)&icrc(t3«'"f>7mmB6.  Hurt 

foltOufat^Mgabfchk^crtaUweegfdc 

<«i»legel24  t&.TnfeOaetfl  1  3  maU4- 

»}n&m4d)t  J  I  X  tt>t>rtrju«&feicr&48  — • 


p!04ff;«vtcHiimcit4i7iH'«'nt>fume 


^fflf« 


Editions.  See  p.  37. 
a  note  in  the  handwriting  of  Prince 
Boncompagni,  together  with  his  colla- 
tion of  the  book.  It  is  interesting  not 
only  for  its  rarity  but  for  the  contem- 
porary coloring  of  the  woodcuts. 

In  the  other  editions  described  the  third  line  of  Fig.  21  reads  3  +  36, 
as  it  should. 


Fig.  21.    From   the 

VVIDMAN 


526 


PRINTia)    ]U)()KS  41 

PETRUS   DE  ALLIACO.     Ed.  pr.  1490.     Augsbur^s  1490. 

Alyaco,  Heliaco,  n'AiLLY.     Horn  in  Compit-gne  in  1350;  chancellor  of 
the  University  of  Paris,  Bishop  of  Cambray,  and  Cardinal.    He  died  in  1420. 

Title.  '  Cocordatia  aftronomic  cu  theologia  //  Cocordatia  af- 
tronomie  cu  hyftorica // narratione.  Et  clucidariu  duol■:{^  prc- 
//cedentium  :  dni  Petri  de  Aliaco  car//dinalis  Camcracenfis.' 
(F.  i,r.) 

Coloplioti.  '  E.xplicit  tractatus  de  cocordia  aftronomice  veri- 
tatis  ;  narrationis  hiftorice//a  dno  Petro  cardinal!  Cameracen. 
completus  in  ciuitate  Bafilieii.  anno  xpi  //  1414  :  menfis.  Maij  die 
decima.'  (P".  33,  r.)  'Opus  concordantie  aftronomie  cum  theo- 
logia  necnon  hyftorica  verita :  nar//ratione  explicit  feliciter. 
Magiftri  Joannis  angeli  viri  peritiffimi  diligeti  cor//rectione. 
Erhardiq5  Ratdolt  mira  imprimendi  arte:  qua  nuper  Venetijs 
nOc  //  Augufte  vindelicorum  excellit  nominatiflimus.  4.  nonas 
Januarij.   1490.'     (F.  55,  v.) 

Description.  4°,  15.6  X  20.4  cm.,  the  text  being  ii.i  x  14.7 
cm.     56  ff.  unnumb.,  39  11.     Augsburg,  1490. 

Editions.    There  was  no  other  edition. 

This  work  has  been  included  in  the  list  because,  while  chiefly  astro- 
nomical, it  throws  considerable  light  upon  the  early  Computi.  It  was 
written  to  show  the  relation  between  theology  and  astronomy,  and  hence 
it  has  an  important  bearing  upon  the  study  of  the  medineval  calendar. 

ALONSO   DELATORE.    Ed.  pr.  1489.  Seville,  1538. 

A  Spanish  savant  of  the  fifteenth  century. 

Title.  '  Vifio  delectable  de  //la  philofophia  ?  ar//tes  liberales: 
meta//phifica  :  y  philofo-//phia  moral  .:.// M.d.xxxviij.'  (This 
is  surrounded  by  an  elaborate  woodcut.)     (F.  i,  r.) 

Colophon.  '  CFenefce  el  libro  llamado  vifion  delectable  dela 
Philofophia  ?  ar//tes  liberales.  Es  impreffo  enla  infigne  y  muy 
leal  ciudad//de  Seuilla  en  cafa  de  Juan  Cr6berger.//Aiio  de 
.M.d.xxxviij.'     (F.  Ixxij,  r.) 

Description.  Fol.,  19  X  26.9  cm.,  the  text  being  15.3  X  24.3 
cm.     72  ff.  numb.,  42  11.     Seville,  1538. 

Editio7is.    Seville,  1489,  fob;  ib.,  1538,  fol.  (here  described). 


42 


RARA   ARITHMETICA 


This  rare  treatise  is  an  encyclopaedia,  with  chapters  devoted  to  the 
various  arts  and  sciences.  The  arithmetic  is  found  in  Chapter  IV 
(see  Fig.  22),  and  consists  of  only  two  pages  of  theoretical  discussion. 

CCapituloau!.lE>daanrinctbicar&cfU9&iueM- 

r^fl',v>Dgfuv?tiiidadomodo:pocmugringularc6fecrct08« 


'HflandopaifltrauciTandoertefendfro:  rmieronencima  oel 
m6tc:3  Dofecome^auarnmarauillolocaminoiel  ^lloeguio 
cnpnlugarrecafaeipalaaojmujfingulflrfst'zalapucrtflrc 

UrillabflllarornamujpragacifTimaimuj^^funda  rojcllaD 

rciencia.iaqualauiiqlosmicbzoscubzienecoabitofeminilrparcfaaoc 
bajro  i5  aql  afcodcr  coza(;6  oc  mu^  penetrate  1  mu^  mgeniofo  ra  rc.r  en 
laoicftrarcniapngrafioiJbierroijenlarmieftrarnatablacbla'qu^ada: 

Fig.  22.    From  the   1538  alonso  delatore 
NICOLAUS  CUSA.     Ed.  pr.  c.  1490.     Strasburg?,  c.  1490. 

NicoLAUs  CusANus,  NicoLAUS  Chrypffs  OR  Krebs.  Bom  at  Kues  on 
the  Mosel  in  1401  ;  died  at  Todi,  Umbria,  August  it,  1464.  He  held  posi- 
tions of  honor  in  the  Church,  including  the  bishopric  of  Brescia.  He  was 
made  a  cardinal  in  1448.    He  wrote  several  other  works  on  mathematics. 

Ttt/t\  Usually  known  as  the.' Opuscula.'  The  work  begins: 
'  Prohemium.//  [I]n  hoc  volumine  ptinentur  certi  tractatus  :  libri 
altiffime  ptemplat6//nis  et  doctrine:  a  preclare  memorie  pre- 
ftantiffimo  doctiffimoq5  viro // Nicolao  de  Cufa.'     (F.  i,  v.) 

Description.  Fol.,  17.5  X  25.4  cm.,  the  text  being  1 1.7  X  18  cm. 
163  ff.  unnumb.  +  i  blank  =  164  ff.,  45  11.  S.  1.  a.  (Strasburg  .?, 
c.  1490.  Some  bibliographers  place  it  as  early  as  1480  and 
others  assign  it  to  Milan  as  late  as   1505.) 


PRINTED    BOOKS  43 

Editions.  Cusa's  'Opuscula  varia'  fkst  appeared  s.  1.  a.  (Slras- 
burg?,  c.  1490  or  earlier).  His  'Opera'  appeared  at  Paris  in 
151 1  and  1 5  14  (see  below),  and  again  at  Basel  in  1565. 

This  contains  fifteen  of  Cusa's  tractati,  including '  Reparatio  kalendar  i  j  ,* 
'De  Apice  theorie'  (4  if.),  'De  mathematicis  complementis,'  'I)e  mathe- 
matica  perfectione.'  Of  these  mathematical  chapters  the  first  two  are 
of  some  interest  in  the  history  of  arithmetic,  the  others  referring  chiefly 
to  mensuration. 

NICOLAUS   CUSA.     Ed.  pr.  c.  1490.  Paris,  15 14. 

See  p.  42. 
Title.    '  Hec  in  hoc  fecudo  vo//lumine  contcnta.// 

Dialogus  de  ignoto.    2.  De  tranfmutationibus 

Dialogus  de  annuncia-  geomctricis.     33. 

tione.     3.  De  Arithmcticis  com- 

ExcitationQ  libri  X.     7.  plenientis.     54. 

Coniectura  de  nouiffi-  De  mathematicis  com- 

mis  diebus.//    i.  plementis.     59. 

Septem  epiftolae.     3.  Complementum  theo- 

Reparatio  Caledarii.    22.  logicum.    92. 

Correctio  Tabularum  De  prefectione  mathe- 

Alphonfi.     29.  matica.     iii.' 

(Woodcut  of  printing-press,  with  the  words:  'Prelu  Afcefianu.') 
'  Venudantur  cum  cete//ris  eius  operibus  in  Acdibus  Afcenfi- 
anis.'     (F.  I,  r.) 

Colophon.  '  Emissvm  est  hoc  librorvm  Cvsae  opvs  //  egregivm 
Parisiis :  ex  officina  Ascen//siana  anno  Christi  pientissimi  om// 
nivm  Redemptoris  MDXIIII,  octa//va  Assvmptionis  semper 
San//ctae  semperqve  Virginis //Christi  Deiqve  Matris//Mariae. 
qva  patroci//nante  apvd  Filivm  //  portvm  salv-//tis  spcra-//mvs 
et//veniae.'    (F.  CXVI,  r.) 

Description.  Fol.,  20x29  cm.,  the  text  being  12.7x26.8 
cm.  114  ff.  numb.  (Roman  numerals)  +  2  unnumb.  =  i  16  ff., 
46  11.     Paris,  1 5  14. 

Editions.    See  above. 


44  RARA   ARITHMETICA 

This  second  volume  of  the  Paris  edition  of  Cusa's  works,  edited  by 
Faber  Stapulensis,  contains  the  '  tractati '  already  mentioned  on  p.  43. 

JOHANN   WIDMAN  (?)     Ed.  pr.  c.  1490.      S.  1.  a.  (c.  1490). 
See  p.  36. 
Title.   'Algorithmus  Integro:^//Cum  Probis  annexis.'    (F.  i,r. 
See  Fig.  23.) 

Description.  8°,  14.9  X  20.6  cm.,  the  text  being  7.8  X  14  cm. 
12  ff.  unmimb.,  29-31  11.     S.  1.  a.  (c.  1490)- 

Editions.  The  various  bibliographies  assign  different  dates, 
but  I  presume  there  was  only  this  one  edition.  De  Morgan  (p. 
99),  whose  judgment  as  to  the  dates  of  such  early  works  was 
unreliable,  estimated  this  as  "hardly  later  than  I475."  adding, 
"I  think  this  is  the  oldest  book  in  my  list."  Wappler,  who  has 
critically  investigated  the  matter  {AbJiandlungen,  V,  158)  believes 
that  Widman  wrote  this  work,  the  'Algorithmus  linealis'  (p.  36), 
and  also  the  'Algorithmus  Minutiarum  Phisicarum.' 

The  work  opens  with  a  quotation  from  Boethius,  the  same  indeed 
as  the  opening  sentence  of  Sacrobosco's  Algorismus.    After  treating 

of  numeration,  addi- 
tion, subtraction,  du- 
plation,  mediation, 
multiplication,  and 
division,  the  author 
takes  up  progressions, 
roots,  and  the  proofs 


Fig.  23.  Title  of  the  .algorithmus  integrorum 

of  the  various  processes.    There  are  no  applications  in  the  book. 

Other  works  of  i4go.  Anianus,  p.  32,  1488  ;  Boethius,  p.  27,  1488. 
There  also  appeared  c.  1490,  at  Leipzig,  an  anonymous  work  edited  by 
Norico,  entitled  *  Arithmeticae  Textus  communis.' 

ANONYMOUS.     Ed.  pr.  c.  1491-  S.  1.  a.  (c.  1491). 

Title.    See  Fig.  24. 

Colophon.  'CFinis  trium  Algorifmo:^  cum  proper //tionum 
vel  Mercatorum  regula.'     (F.  10,  r.) 

Description.  4°,  13.7  X  19.4cm.,  the  text  being  8.6  X  15.6cm. 
10  ff.  unnumb..  34-36  11.     S.  1.  a.  (c.  1490- 


PRIN'ri':i)    1U)()KS  45 

Editions.  This  is  not  the  edition  dcscribctl  1)\-  Dc  Morgan 
(p.  99),  because  the  title  has  '  addita  etiam  regula,'  instead  of 
'  addita  regula.'     The  word  'regula'  instead  of  '  regla '  at  the 

jOfUgozifitms  nouusoe 

^  *-*»intcgn8 copcndforcfmcfiGUrarumC mo:c ^fwlo 
rum|Dclf  nonccompilatU0»  grtem  numcrandi  omncrtKa 
vwiticalculandi  ciHJcliatim  bzcuinlmc  cdoccno.vna  cu  ai 
0o:irmi8  DC  mmurijsvuIgaKbua  videlicet  crpbificalibua 
Sddiia  ctiam  rcgula.|po:tionu  tarn  DcmrcGns  qpfracns 
qacrutgo  mcrcaro2om  regulatJiaf  Quibtm'babine.  <!iuio 
niodtca  adbibirdDiUgetmaomncmcaUuUndttnodum  fa 
dUtme  adiptfapoceil 


■»■*«)»  ■»♦«•/- 


6/     '?EN,tift/pIfcl 

j    ^"^       j£tfifOiuifo:rcjfipeP     IZ    ff   4 


Fig.  24.    TiTLK   pa(;k  ok  thk  al(;okismus  of  c.  1491 

end  shows  it  to  be  different  from  the  one  described  by  Brunet 
(edition  of  i860).  It  is  probably  no.  827  of  Hain,  and  the  inter- 
nal evidence  makes  1491  the  probable  year  of  the  c-omi)osition 


46 


KARA   ARITHMETICA 


or  publication.  De  Morgan  says  that  three  editions  are  known, 
and  Giinther  on  the  authority  of  Chasles  gives  an  edition  at 
Cologne,  c.  1510. 

This  anonymous  work  is  of  the  class  of  the  arithmetics  of  Muris, 
Peurbach,  Ciruelo,  and  other  mediaeval  and  early  Renaissance  writers. 


picrngorns  aritbmetn'reinrroducfor 


Fig.  25.   TnLE  page  of  calandri 


It  contains  a  brief  treatment  of  the  '  species  '  (the  fundamental  opera- 
tions) with  integers,  omitting  '  duplatio  '  and  '  mediatio  '  but  including 
progression  and  roots.  This  is  followed  by  a  discussion  of  common 
fractions  (' Algorifmus  nouus  de//minutijs  vulgaribus '),  a  single  page 
on  sexagesimal  fractions  ('de  minutijs  Phificalibus '),  and  a  page  on 
proportion.  It  is  one  of  the  first  books  to  identify  proportion  with  the 
rule  of  three,  or  merchants'  rule  as  it  was  often  called.  ('  De  regula 
proportionum // Sive  aliter  Regula  Mercatorum  dicta.')  It  is  not  as 
practical  as  the  elaborate  title  would  seem  to  indicate. 


PRINTED    BOOKS 


47 

Florence,  1491. 


PIIILIPPI    CALANDRL     Kcl.  pr.  1491. 

A  Florentine  arithmetician  of  the  fifteenth  century. 
Title.    '  Pictagoras    arithmetrice    introductor.'     (Woodcut   of 
^3 


8] 


•parri 

r?4^>pcr 
00O44- 

8? 

Uicnttc 

4-r 

f?4. 

1^8 

4f 
0 

4r 

l_?i 


•Parti  1  E 

<^o         -parti  r?>:t  E    '"^ 

i-Co 

i]>^  —  i-^ 

0  \/-h 

0     -^h 

Uicttnc  1  1    a 

nicmc  tjtt 

Co  — i 

I          -Parti  >  e  1 
>  — 1 

480 

mcttjic    1  (^0 

i. 

n            ?!/>            |> 

Uicntie    0     fji 

Fig.  26.    From  calandki's  akithmktic 

Pythagoras.)  (F.  i,  v.  Fig.  25.)  '  Philippi  Calandri  ad  nobi- 
lem  et  ftudiofu,^  Julia//num  Laurcntii  Mediae  dc  arimethrica 
opufciilu.'    (F.  2,  r.) 


48 


KARA   ARITHMETICA 


aticoTidoctoempieii 
nafontcm'vdtrqu.i 
do  c  picna  lion  merren 
do  ilco'.\<*.octo  1  ftur.-i 
doiluoratotofluorcrc 
beladccrafoittcin  i  i 
di :  do  fripere  elTertdo 
uora  la  fo;ir:  -rmccrcn 
do  i!f3ndp«o  T  flura 
doihioraToiotfiqu.iu 
di  iftva  plena  la  dcaa 
fonte 


Qnorerpeiriccin  u.ii)' 
rottdodiiii  po>o  cl^e  el 
fidrciiro  r o  l>:ai:cid ;  ct  i 
uofendouTfti-t*  J^ioraol 
gnidi  falc-^  dibrarno! 
crdtpoi  lanocre fcmd: 
3  dibx^do :  uo  faperc  , 
in  ciuari  dt  fara  fnora',' 
dcldecropeNO 


IM 


J      <:? 


!  farafuora  !  r?>  di 


Fig.  27.    From  calandri's  arithmetic 

Colophon.    'Impreffo  nella  excelfa  cipta  di  Firenze  per  jj  1 1 
Lorenzo  de  Morgiani  et  Giouanni  //  Thedefco  da  Maganza  fi// 

nito  a  di  primo  di  //  Genaio 
1491.'  (F.  104,  V.  F'ig.  30.) 
Description .  8°, 9. 8x13. 2 
cm.,  printed  in  double  col- 
umns, each  3.3  X  10.7  cm.  to 
1 1.5  cm.  104  ff.  unnumb., 
9-26  11.  Florence,  149 1. 
Editions.  Florence,  1491, 


Fin.  28.  From  calandris  arithmetic 

8°  (here  described)  ;  ib.,  i  5  18,  printed  by  Bernardo  Zuchetta,  4 


PRINTKI)    HOOKS 


49 


Of  this  rare  book,  the  first  in  De  Morgan's  list,  Mr.  Plimi)ton  ])()S- 
sesses  two  copies.  It  is  beautifully  printed,  and  is  practical  in  its 
presentation  of  the  operations,  but  traditional  in  its  problems.    It  is 


IS 


Eglicuna  torrccbcc 
flUa  +  obrafdaTdflp 
picmpail'atmofuimc 
cbcclarglx)  Jo  bine 
da.  uo  fapcrc  qnamo 
fara  htiujba  una  ftinc 
cbcfiaaipfncataallnn 
ua  ddnumc  tallad 
maslclUtorrc 


-}o 


90( 


laradiccdi^^f  oo 
faralunja     io  brae 


Eglicuiialbcroiiifula 
niiadunfuimc  clqua 
Ic  c  aire ;  o  brdccia  cl 
fi«mccIargbo)obra 
dazpcrfortiinadiiic 
tofiruppjiiiMl  luogo 
cbclacima  dcllalbcro 
toccaiui  Lariua  dclfm 
iDcGorapcicquaiirc 
bracaa  fcnc  nippc  t 
qnanto  ttcrimafc  rino 

fo 30 

^\-fo  ■'-roo  ^jo 

I  oo|         900 


riinafcrito  iC  brae 

da  z  }  4  bracda  fcnc 

j  ruppe 


Fig.  29.    From  calandki's  arithmetic 


the  first  printed  Italian  arithmetic  with  illustrations  accompanying  prob- 
lems, and  the  first  to  give  long  division  in  the  modern  form  (Fig.  26) 
known  to  the  Italian  writers  by  the  name 
'  a  danda.'  Indeed  Calandri  gives  only  the 
'a  danda'  method,  omitting  the  galley 
form,  and  is  therefore  fully  a  century  ahead 
of  his  time.  De  Morgan's  statement  that 
he  uses  a  divisor  diminished  by  i  is  incor- 
rect, as  will  be  seen  from  Fig.  26.    Figs.  27 

and  29  show  that  the  problems  of  the  cistern,  the  snail  (serpent)  in  the 
well,  the  length  of  the  hypotenuse,  and  the  broken  tree  were  familiar 
in  Calandri's  time. 

0//u-r  uwrks  0/1491.    Anianus,  p.  32,  1488  ;  Boethius,  p.  27,  1488  ; 
Borghi,  p.  16,  1484. 


^niprcdb  iiclla  nrcira  tipta  di  ^veayc.  per  J 

ioi  CM  >o  di-  BJorgtnni  ct  tiiout 

Xi5i;ddi:o  da  iDijaiija  ft 

niton  dipnmodi 

0cniioi4ji 


Colophon  of 
calandri 


Fu;.  30 


50  KARA   ARITHMETICA 

FRANCESCO    PELLOS  or  PELLIZZATI. 

Ed.  pr.  1492.  Turin,  1492. 

A  native  of  Nice,  living  in  the  latter  half  of  the  fifteenth  century. 

Title.  '  Sen  fegue  de  la  art  de  arithme-/Aicha.  et  femblat- 
ment  de  ieume-//tria  dich  ho  nominat5  Copendio  //  de  lo 
abaco.//i  234567890.'     (F.  i,  r.    Fig.  31.) 

Colophon.  '  Complida  es  la  opera,  ordinada.  he  condida  //  Per 
noble  Frances  pellos.  Citadin  es  de  Nifa.  .  .  .  Impreffo  in 
Thaurino  lo  prefent  copendio  de  abaco  per  mei/  //ftro  Nicolo 
benedeti  he  meiftro  Jacobino  fuigo  de  fancto  ger//mano.  Nel 
anno  .1492.  ad.  Di  .29.  de  feptembrio.'     (P\  80,  r.) 

Description.  Sm.  4°,  13.8  X  20.9  cm.,  the  text  being 
9.2  X  15.2  cm.  80  ff.  numb.,  39  11.  Some  of  the  initials  have 
been  in.serted  by  hand,  in  red.     Turin,  1492. 

Editions.    There  was  no  other  edition. 

This  is  one  of  the  rarest  arithmetics  known  to  exist.  (Brunet,  IV, 
475;  Graesse,  Tresor,  V,  100;  Riccardi,  I,  2,  256.)  The  only  good 
description  of  the  work  is  that  given  by  Boncompagni  in  the  Atti 
deir  Accad.  Pontif.  de"  nuovi  Lincei,  XVI,  161,  332,  evidently  after 
examining  this  copy,  since  it  bears  a  note  in  his  handwriting. 

Pellos  first  considers  the  fundamental  operations  with  integers,  fol- 
lowing this  by  a  treatment  of  proportion,  square  root,  and  cube  root. 
He  then  discusses  the  subject  of  fractions  in  much  the  same  order,  the 
rule  of  three,  certain  rules  relating  to  weights,  time,  money,  and  other 
measures,  and  such  topics  as  partnership,  barter,  interest,  alloys,  and 
the  rule  of  false  position,  single  and  double.  He  closes  the  work  with 
a  chapter  on  mensuration,  or  as  he  calls  it,  '  De  la  art  de  ieumentria ' 
('  ieumetria'  in  the  title),  and  gives  a  number  of  interesting  woodcuts. 
The  chief  interest  of  the  book  attaches,  however,  to  the  fact  that 
Pellos  came  very  near  the  invention  of  decimal  fractions,  and  that 
he  actually  used  the  decimal  point  as  is  shown  in  the  illustration 
(Fig.  32).  It  cannot  be  said,  however,  that  he  had  any  conception 
of  the  real  value  of  the  decimal  fraction  as  such,  the  first  book 
devoted  to  this  subject  being  'La  Disme  '  of  Stevin  (1585),  hereafter 
described.  Pellos  simply  uses  the  decimal  point  to  indicate  division 
by  some  power  of  ten,  writing  a  common  fraction  in  the  quotient. 
Thus,  to  divide  425  by  70,  Pellos  would  divide  42.5  by  7,  writing  the 
result  6 -■';,. 


PRINTHl)    HOOKS 


51 


Fig.  31.    Title  page  of  pellos 


52                    RARA   ARITHMETICA 

ri 

fTPdrrfrpcr              2     ol 

7     9<^J4.8?9 

♦  7 

qnoctcnt       $9^27419 

C'parnVpcr            5    0 

^    8    ?     «$     0     4.  .? 

1  7 

2  0 

quocicrtt       1     9     4     S    6    4, 
IT'Pflrtirpfr            7     0 

0      ?      ?      T      9      1   *9 

2    ; 
$    0 

qoodcnc       (     ;     ^    3    S    S 

6  9 

7  0 

CP^rtiVpcr          I   0   0 

^     9     7    6     S  .9     7 

quocicnc 

CP^rrfrpcr          4    0     oj 

7     S     9     6     $  ,7     ^ 

quocicnt      1^74* 

»75 

400 

Cr*P«t-{irpcr        3000 

8     7     6     5     8  .7    9_ 
quocient      29219 

1 

179  i 

3000 

(L^un  dqiU  manicrjipodce  fdfrc  Ibc  (one  fcmbUdpanimco 

Fic.  32.    From  the  peli.os  arithmetic 


\  pdrair 


PRINTED    BOOKS  53 

GEORG  VON    PEURBACH. 

Ed.  pr.  1492.  Wittenberg,  1534. 

PuRBACH,  Peuerbach.  Bom  at  Teuerbach,  Upper  Austria,  May  30,  1423; 
died  at  Vienna,  April  8,  1461.  He  studied  under  Johann  von  Omiinden  (see 
p.  117),  Nicolaus  Cusa  (see  p.  42),  and  other  great  teachers,  and  later 
he  became  professor  of  mathematics  at  Vienna,  where  Regiomontanus 
(Johannes  Miiller,  of  Kiinigsberg)  was  his  pupil.  His  interests  were  almost 
entirely  in  astronomy. 

Title.  '  Elemen//ta  Arithmetices  //  Algorithmvs  de  //  nu- 
meris  integris  auctore  //  Georgio  Peurbachio.//  De  Nvnieri.s 
Practis,//  Regulis  communibus  &  //  Proporcionibus.//  Cum  prae- 
fatione  Philippi // Melanchthonis.// M.  D.  XXXIIII.'    (F.  i,  r.) 

ColopJion.  '  Impress vm  Vitebergae // per  losephvm  Clvg.// 
Anno  M.  D.  XXXIIII.'     (F.  39,  v.) 

Description.  8°,  10.8  X  15.6  cm.,  the  te.xt  being  6.4  X  11.2 
cm.     39  ff.  unnumb.  +  i  blank  =40  ff.,  22-25  ^1-    Wittenberg, 

1534- 

Editions.  S.  1.,  '  Explicitum  est  hoc  opus  anno  Christi  dom. 
1492.'  4°;  Vienna,  s.  a.,  c.  1500;  Leipzig,  1503,  4°;  ib.,  1507, 
4°,  and  probably  ib.,  15  10  and  ib.,  151 1;  Vienna,  s.  a.  but  ear- 
lier than  i  5  1 1,  4° ;  ib.,  1511,4°;  ib.,  1512;  Niirnberg,  1513  ; 
Vienna,  15 15,  4°;  ib.,  1520,  4°;  Wittenberg,  1534,  8°  (here 
described);  ib.,  1536,  8";  ib.,  1538;  Venice,  1539,  8°  (see 
below);    Frankfort,  1544. 

The  arithmetic  of  Peurbach  went  by  various  names,  as  '  Opus  Algo- 
rithmi,'  '  Institutiones  in  arithmeticam,'  '  Elementa  arithmetices,'  and 
'  Introductorium  in  arithmeticam.'  It  is  a  brief  treatise  on  the  funda- 
mental operations  with  integers  and  fractions,  and  contains  a  few  simple 
applications.  Peurbach  was  too  profound  a  mathematician  to  have  con- 
sidered it  a  work  of  any  importance,  but  it  is  probable  that  he  wrote  it 
for  the  benefit  of  students  who  were  not  yet  prepared  to  take  up  his 
work  in  astronomy. 

GEORG  VON    PEURBACH.    Ed.  pr.  1492.      Venice,  1539. 

See  above. 

Title.  '  Elementa  //  Geometriae  ex  Evclide  //  fmgulari  pru- 
dentia  collecta  a  loane   Vo-//gelin   profeffore   Mathematico  in 


54  RARA   ARITHMETICA 

//fchola  Viennenil.// Arithmeticas  practicae  per  Georgium // 
Peurbuchium  Mathematicum.//  Cum  praefacione  Philippi  // 
Melanchthonis.'  (Woodcut  with  motto  :  '  Dissimilivm.  Infida. 
Societas.')  (F.  i,  r.)  The  arithmetic  of  Peurbach  begins  on  f. 
32:  '  Elementa//Arithmetices.//Algorithmvs  de  nv-//meris 
integris,  fractis,  ReguHs  //  communibus,  &  de  Pro-//portionibus. 
//Authore  Georgio  Peurbachio.// Omnia  recens  in  kicem  aedita 
fide  & //diligentia  fmgulari.//Cum  praefatione  PhiHp.  Melanth.' 

Colophon.  '  Venetijs  loan.  Anto.  de  Nicohnis  de  Sabio.// 
Sumptu  uero  D.  Melchioris  Seffae.  Anno  //  Domini  M  D 
XXXVIIII.//Menfe  lanuario.'     (F.  6^ ,  v.) 

Description.  8°,  10.3  X  15.8  cm.,  the  text  being  8  X  12.2  cm. 
I  f.  unnumb.  +  68  numb.  =69  ff.,  29  11.     Venice,  1539. 

See  p.  53. 

Other  wo7-ks  of  I4g2.    Anianus,  p.  32,  1488;  Boethius,  p.  27,  1488. 
Works  of  i4gj.    Anianus,  p.  32,  1488;  Anonymous  (see  Widman), 
p.  36,  c.  1490;  Isidorus,  p.  8,  1483;  Vincent  de  Beauvais,  p,  10. 

LUCA   PACIUOLO,   DE   BORGO   SAN    SEPOLCRO. 

Ed.  pr.  1494.  Venice,  1494. 

Paciolus,  Patiulus,  Pacioli.  Bom  in  Borgo  San  Sepolcro,  Tuscany,  c. 
1445-1450;  died  soon  after  1509.  Not  an  original  mathematician,  but  the 
compiler  of  several  works. 

Title.    See  Fig.  33. 

Colophon  to  the  part  on  arithmetic :  '  Et  fi  fequenti  pti 
pncipali  Geo",  finis  decima  nouembris  ipofitus  fuerit :  huic  tamen 
pti:  die  vigefi//maeiufdem  Ipofitus  fuit.  M'.'cccc.lxliiij.  Per  eof- 
dem  correctorem  1  imprefforem  vt  i  fine  Geo^  hr.'  (F.  232, 
numb.  224,  V.)  There  is  also  the  following  date  on  f.  i,  v., 
'  M°ccc.?lxliiii?xx^.  Nouembris.' 

Description.  Fol.,  21.5  X  30.5  cm.,  the  text  being  19  X  24.2 
cm.  8  ff.  unnumb.  +  224  numb.  =232  ff.  in  the  part  on  arith- 
metic ;  76  ff.  numb,  in  the  part  on  geometry  ;  making  a  total  of 
308  ff.,  56-60  11.     Venice,  1494. 

Editions.  Venice,  1494,  fol.  (here  described);  Toscolano,  1523, 
fol.  (p.  58). 


PRINTED    BOOKS  55 

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metmlC>:opo:tioni^  p:o/ 
poztionalita, 

iConrincntiaOctuttalopcra. 

fee  numm'c  mifurc  in  turn"  modioccurrcntt. 
^:opo:rionic^pom63lit3anotin3  ocl.s.  oe£ucii 

dceoctuttilialmTonto'  ,  -  ....««, 

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belaregola  mcrcantefca  Oitta  cd/,^  foi  fodameit; 

t!Concaricrcniplaripcrcrm?S.@^uadafimHM:rdi 

S"tiWplicar.fummar.cfotraroetcpjcpo;[io 

nicoctutKfojriradici.   ,  ., 

be  k.vrcaolc  ocl  catayn  T>m  pofinoc  cfua  o^!gie. 
£u  Sgcncraliouerconclufiom- nJ66.abfdua^ 

cS(bcfxperrc60lc.o:dinarieno,fi  podelTe, 
Sfo2KbmomiicrcdficaltrclmceuTan63{iDcC 

Scrc^coealgcb:ae(ttctxlacofa  do:  fab2i' 
dxcfondamcnri. 

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^odde«bcftiamf.clo2partirc  ,  ._  ..  ^^. 
^itti:pcfd6i:cotrimi:ttueUi:logagiom:csodfm«itT. 
5fearam'  i  tuKimodi  fcmplid-.conipolb-.c  cd  tempo. 
iCambi  Kali.fcccbi.fitritii^  i>f  minuri  ouer  comunU 
0sn(i  femplici  c  a  capo  wnno  e  altn  termini. 
lufti.f3Wi.fc0nti.dc  tempo  cocnanela  rccarc  aun 

dipiu  partite*  ^ 

e2t4r«cti'.elo20affinarc.ecarattar6  ^      . 

flEK)Wcafieragioni  ftrao2dinane  vane  c  ©luerie  8 
tutteoccurcmiccommondlafcquente  tauola  ap/ 
pareoidinatamenteoetutte.  .^r^..^ 

ejdineafapcrtenerogmcotocfcnpmreccclqua 

^S  oe  wttcvfan^C  C  coftumi  mcrcantcfcbi  in  tut 

^^^an'ca  e  rixo2ica  oc  geometria  c  t)cli.^co^pi  regii^ 

i^?Ju1fflf",?and«ri^^^^ 
mo  oifufamcntepcr  la  fcqucntc  tauola  appare. 

Fig.  33.   TiTLK  i-agk  of  the   i494  paciuolo 


56 


RARA   ARITHMETICA 


This  volume,  the  first  great  general  work  on  mathematics  printed, 
includes  treatises  on  arithmetic,  algebra,  and  geometry,  each  being 
considered  from  a  somewhat  scientific  rather  than  practical  standpoint. 
The  arithmetic,  for  example,  gives  the  various  methods  in  multiplication 
(see  Fig.  34)  and  division,  instead  of  emphasizing  the  one  or  two  most 
prominent  in  business  circles.    In  the  same  way  Paciuolo's  treatment  of 


XXX 


9  7  4  I  6  9 

Fig.  34.    Gelosia  multiplication,   1494  paciuolo 


the  rule  of  three,  the  rule  of  false  ('El  cataym'),  partnership,  pasturage, 
barter,  exchange,  and  interest,  while  nominally  practical,  was  too  elabo- 
rate for  the  mercantile  schools.  His  was  the  first  printed  work  to  illus- 
trate the  finger  symbolism  of  number  (Fig.  35).  Paciuolo  copies  without 
hesitation,  practically  verbatim,  from  the  work  of  Chiarini  (p.  10),  and 
doubtless  laid  under  contribution,  after  the  manner  of  his  time,  various 
other  works  of  his  predecessors.  In  algebra  he  used  the  common  sym- 
bolism of  the  time  for  the  unknown  quantities  and  for  roots,  but  he  made 
use  of  no  symbols  of  operation.  This  part  of  the  treatise  relates  chiefly 
to  surd  numbers.  In  geometry  he  follows  Euclid's  Book  I  very  closely, 
but  departs  quite  radically  from  the  subsequent  books.  The  work  had 
a  great  influence  on  subsequent  writers,  including  the  English  Tonstall 
(p.  132).  Paciuolo's  training  had  fitted  him  to  write  a  treatise  of  this 
nature.  He  had  been  a  tutor  in  the  family  of  a  Venetian  merchant, 
had  traveled  extensively,  had  come  in  contact  with  practical  mathe- 
maticians, and  had  studied  the  ancient  mathematics  in  the  cloisters  ;  and 
traces  of  all  these  influences  are  seen  in  his  work.  In  1497  Paciuolo 
wrote  at  Milan  a  work  entitled  '  Divina  proportione,'  which  was  pub- 
lished at  Venice  in  1509  (p.  87).  He  also  published  an  edition  of 
Euclid  at  Venice  in  1509. 

Other  works  0/14^4.    Albert  of  Saxony,  p.  9,   1478;  Anianus,  p. 
32,  1488. 


PRINIEU    HOOKS 


57 


he  ['"'^'"''^'^-•f^^  vi-JtVJ'/' 


A-W^i..-  j/-*^-  If  ■bj^rrt-/ 


jfry./hc'- 


Biflinctio  rccunda.Zracfaf  ofl  quartiie. 


T)^^(!- 


35.     P'lNHER    SYMBOLISM,     1 494    PACIUOI.' 


UNI 


58  RARA   ARITHMETICA 

LUCA   PACIUOLO,   DE   BORGO   SAN    SEPOLCRO. 

Ed.  pr.  1494.  Toscolano,  1523. 

Title.  '  Summa  de  //  Arithmetica  geo//metria.  Proportioni : 
et  proportionalita  :  //  Nouamente  impresfa  In  Tofcolano  fu  la 
riua  dil  Benacenfe  et  //  vnico  carpionifta  Laco :  Amenisfimo 
Sito :  de  li  antique  2  //  euidenti  ruine  di  la  nobil  cita  Benaco  ditta 
illuftra-//to :  Cum  numerofita  de  Impatorij  epithaphij/Zdi  antique 
:  perfette  littere  fculpiti  do-//tato:  2  CU5  finisfimi  2  mirabil  co-// 
lone  marmorei:  inumeri // fragmenti  di  alaba-//ftro  porphidi  2 
ferpentini.  Cofe  certo  //  letter  mio  diletto  oculata  fi-//de  miratu 
digne  fot-/Aerra  fe  ritro//uano.//  Continentia  de  tutta  lopera:'// 
(The  rest  of  the  title  page  is  substantially  identical  with  that  of 
the  1494  edition.)    (F.  i,  r.) 

Colophon  to  the  part  on  arithmetic  :  '  CEt  fi  fequeti  pti  pnci- 
pali  Geo^  finis  decima  nouembris  ipofitus  fuerit:  huic  tamen  pti 
die  vigefima//eiusde  impofitus  fuit  .MVccccc.xxiij.  Per  eosdem 
correctors  impresforem  vt  in  fine  Geo^  hetur.'     (F.  232,  v.). 

Description.  Fol.,  21.  i  X  30.4  cm.,  the  text  being  18.2  X  23.5 
cm.     9  ff.  unnumb.  +  223  numb.  =  232  ff.     Toscolano,  1523. 

Bound  with  this  is  the  '  Tractatus  Geometria.  Pars  fecunda 
principalis  huius  operis  2,  primo  eius  diuifio.'  This  part  of  the 
book  contains  75  ff.,  besides  the  index. 

Editions.  See  p.  54.  De  Morgan  has  shown  (p.  2),  that 
there  are  slight  differences  between  the  copies  printed  in  1523, 
proving  that  a  second  impression  was  necessary  in  that  year. 

See  p.  56,  and  Fig.  36. 

PEDRO   SAnCHEZ   CIRUELO. 
Ed.  pr.  1495.  Paris,  1505. 

Born  at  Daroca  in  Aragon,  c.  1470 ;  died  at  Salamanca  in  1 560.    One  of  the 
most  learned  men  of  his  time.    He  was  professor  of  philosophy  at  Alcala. 

Title.  '  Tractatus  Arithmethice  //  Practice  qui  dicitur  //  Algo- 
rifmus.'  (A  large  woodcut  with  the  initials  D.  R.  and  the 
inscription  '  Alaventvre.  Tout  Vient.  Aponit.  Qvi.  Pevt.  Atendre. 
Denis.   Roce.')     (F.  i,  r.) 


PRINTED    BOOKS 


59 


adillufhiffimC^iincipcmSm-abalduj  Urhini 

Bucrm  tl^ofUio  ftrrtri:*  DurantiaEomiimi.  Orfds  Ulin(f<5  litfcria  ©matifR' 
in«mw«l^il?emimaoi*ip(uu:Qito;cnfrruaUiffminm;fr8matwePt)8urgo  ] 

0quantita  ^asnanimo  ©uca:c 

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je>CTxt>:  iTJnna  tDgiujfaate  frr  w?  mcdo  alcbana  a>la 
in  rcmm  nattra  fema  In  no  potue  mrtcrc  dicr  taqual 
0)6  tt  lo  intmdo  Cosn  ladiii)  tr  atoi  cfjc  li  noftri  Itnft 

lofopbi  nonnt  fta  cnpiOfemenK  tractato;  e  in  Ztxoncii  c 
pacnca-  dha  p»rd?t  Icsoica  sia  alii  tempi  noftn  fonno 
moltD  ob(tun.e  wi  mota  miit  apsSi :  c  aOt  pMcncfx  nil 
gun  malf  appUcaB :  tiilcftf  m  tao  opranora  moltD  fa' 

riano:  c  crn  uradi  c  labraofi  affimra  mauno  in  oprra:  fi 

« .uiiuu  UoniOBmlfiire:  rndm  in  pirtando  ncn  intmdo  (>  nwi  qumtodxsto 
Rama  t  ovmit  fit  mcfficro:  mefcotadod  fcoando  i  taogfjt  opsSunl  and?oia  la  tt^  j 

£v?e  tZ-pri^£  alio:,  pi  fegaim  qudlo  dx  anltctif  CUB  m  tondo  pfta w 
lara.  Zmc  ojim  nujime  fomr  aliqtiid  cum  f^atmir  (uum  quuJ  m  zc. 

freiffinitioneo/cl  oinfto  rierrtt/et  continne  qwrniMflB. 

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3(D  adona.  ta  ^IS  iT^  .jii  uVohm^  copulate  e  giontc  a  ocrto  tmiUnf  com' 
^'S'^'foTutoifsnt-frnaetotc    wcifcremoueTBmmttnnmcro 
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^S^iriTa  ttU  wftraaTda  cri  namcro:  e  pi  ce  ta  osntam 
iBaaomttntoaliwtacfciarattnttttaciarfmo. 

fljBiffimtiotioraeripjopai(fitTta.    2rticDlua  ftcundnff. 

nemow  c  Crecondo  Mdhmo  pf^lofopfpantf )  rna  motolndint  tt  vnte  c5' 
tofta:  z  cffa  vmiia  non  c  nnmero;  ma  ten  pmidpio  a  aaTcfwionnnuro:  c  a 
quelta  meduntJ  laqle  ogra  cDla  e  oicQ  eflcre  vnaiE  ff  cwido  feucnno  iSoctio 
In  (iia  muftca-  e  U  vraia  dafcfjun  numcro  in  ptenlia:  z  pafhun  ta  fua  antftmctia  re- 
^  e  (bndammlD  wii  rmmcnj  lapUa.  Jjqual  piij  nugruficandola  m  l£  cufc  natu- 
ralT oiffi:  m  qucUo  cfjc  fe  ce  f  ratatt  z  vno.  £>mnt  qo  cit:  idco  cll:  quia  rmim  numc- 
mrit  £i|£ancboiacInumcrDminfiranmembiiomifo:pr  qudlo  (tx  uTo  anrtooli 
Sff  d^^>?"<lintirammeft:nnmcra8  tft.lEpr  latcrw  ptiDonectl  (effumrc 
tf ucUde;  U  fiu  few  in  inliralD  ptcre  p.-occdcre:  t  quocuq;  numon  rjto:  can  ptril 
mji'  V  rnitatcm  addcndo .  dlia  noi  pisUaremD  qucUe  parte  a  noi  pui  note  e  aaomo 
^e  i£«ro  oico  con  glialtn  aldjiino  elTcre  pamo:  e^  qucUo  ctx  folo  Da  la  ^mta  e  nu 
mcratee non Fja  altro nnmero: cl?e  intcgralmentt spnto to parta.  aitro c omo co- 
iTifto- et  E  ontOo  dje  w  altro  numero  e'mcfuratD:  ouiTO  numcnato.  £jcmf<um  p;inu 
CsWvr.  1 1.  U  -c-  .7-«.!£wlu  feciidi.£6mo.4a?el  tDi  to  mrfura  c  mimon.-.b. 
^  1  c+lEI'  "•  M-  iS.e  ftmia.iutti  fonno  cira  nuniai  compfti;  no  fot^ftc  colliiia 


Fig.  36.    First  page  of  text,   1523  paciuoi.o 


6o  RARA   ARITHMETICA 

Colopiion.  '  Arithmetice  practice  feu  Algorifmi  tractatus  a 
Petro  fanchez  //  Ciruelo  nouiter  compilatus  Explicit  Impref- 
fus  Parifius  In  //  Bellouifu.  Anno  dni,  1505.  Die  .29.  Aprilis.' 
(F.  14,  r.) 

Description.  4°,  13.2  X  18.4  cm.,  the  text  being  9.7  X  14  cm. 
14  ff.  unnumb.,  35-39  11.     Paris,  1505. 

Editions.  Paris,  1495,4°;  ib.,  1505,  4°  (here  described) ;  ib., 
1509,  4°;  ib.,  1 5 13,  4°  (see  below).  Ciruelo  also  wrote  a 
'  Cursus  quattuor  mathematicarvm  artiu  liberahu,'  Paris,  15 16; 
ib.,  1523;  ib.,  1526;  ib.,  1528;  Alcala,  1516,  fol.  ;  ib.,  15 18; 
ib.,  1523,  fol.  ;  ib.,  1526;  ib.,  1528.  He  also  edited  Brad- 
wardin's  arithmetic,  Paris,    1495;  ib.,    1502. 

Ciruelo  treats  very  briefly  of  the  fundamental  operations  with  integers, 
common  fractions,  and  denominate  numbers.  Following  the  Spanish 
custom  he  uses  cuento  for  million.  There  is  little  that  is  noteworthy  in 
the  book,  and,  like  Peurbach,  Ciruelo  could  not  have  taken  his  contri- 
bution to  algorism  very  seriously. 

Other  rvorks  of  149^.  Anianus,  c.  1495,  p.  32,  1488.  There  also 
appeared  in  1495  the  treatise  of  Herodianus,  '  De  notis  Graecorum 
Arithmeticis  Grsece,'  Venice.  Of  this  work  there  was  an  edition  in 
1525,  and  a  Latin  edition  published  at  Basel  in  1600,  but  it  hardly 
deserves  to  be  classed  as  an  arithmetic. 

PEDRO  SANCHEZ  CIRUELO.   Ed.  pr.  1495.    Paris,  15 13. 

See  p.  58. 

ntlc.  '  Tractatus  arithmetice  //  Practice  qui  dicitur  algorif- 
mus.//  (A  large  woodcut  with  the  name  Jehanlambert.)  Venun- 
dantur  Parrhifijs  a  Johaue  La//berto  eiufdem  ciuitatis  biblio- 
pola  in  fte-//mate  diui  claudij  manente  iuxta  gymna-//fium 
coquereti.'     (F.  i,  r.) 

Colophon.  '  CArithmetice  practice  feu  Algorifmi  tractatus  a 
Petro  fan//che  Ciruelo  nouiter  compilatus  Explicit  Impreffus 
Parifius //per  Anthonium  Auffourt.  Pro  Johane  Lamberto 
eiufdem  ci-//uitatis  bibliopola  in  ftemmata  diui  claudij  manente 
iuxta  gymV/nafium  coquereti.  Anno  drii  .1513.  Dievero  .21. 
menfis  Martij.'     {¥ .  12,  r.) 


PRINTED    HOOKS  6i 

Description.  4°,  13.1  X  19  cm.,  the  text  bcin^^  9.6  x  14. H  cm. 
12  ff.  unnumb.,  43-46  11.     Paris,  15  13. 

Editions.  This  is  the  fourth  edition  (see  j).  60),  and  is  an 
exact  reprint  of  the  1505  copy,  save  as  to  spelling  and  pagina- 
tion. As  a  piece  of  typography  it  is  much  better  than  its 
predecessors. 

THOMAS   BRADWARDIN.    Ed.  pr.  1495.       Paris,  c.  i  5  10. 

Bragwardine,  Brandnardinus,  Breowardyn,  Bradwardyn,  de 
BRAinvARDlNA,  DE  Bredwardina.  Born  at  Hertfield  (Hartfield)  in  the 
diocese  of  Chichester,  c.  1290;  died  at  Lambeth,  August  26,  1349.  On 
account  of  his  learning  he  was  called  'Doctor  profundus.'  He  was  pro- 
fessor of  theology  at  O.xford,  and  died  as  Archbishop  of  Canterbury.  He 
wrote  four  works  on  mathematics. 

Title.  '  Arithmetica  thome  brauardini.// Olivier  Senant // 
Venum  exponuntur  ab  Oliuiario  fenant  in  vico  diui  Jacobi  fub 
si-//gno  beate  Barbare  fedente.'     (F.  i,  r.) 

Colophon.  '  Explicit  arithmetica  fpeculatiua  thoe  brauardini 
bri  re-/Aiifa  et  correcta  a  Petro  fanchez  Ciruelo  aragonenfi 
ma//thematicas  legete  Parifius,  Ipressa  p  Thoma  anguelart.' 
(F.  6,  V.) 

Description.  4°,  20.3  X  27.8  cm.,  printed  in  double  columns, 
each  being  6.8  x  19.8  cm.     6  ff.  unnumb.,  61  11.    Paris,  c.  15  10. 

Editions.  Paris,  1495,  4°;  ib.,  1496,  fol.  ;  ib.,  1498,  4°;  ib., 
s.  a.  (c.  1500);  ib.,  1502,  4°;  Valencia,  1503,  fol.;  Paris,  1504; 
ib.,  1505,4°;  ib.,  s.  a.  (c.  15  10,  here  described) ;  ib.,  15  12;  ib., 
1530;  Wittenberg,  1534,  8°;  ib.,  1536,  8°.  His  '  Tractatus  de 
proportionibus '  appeared  in  several  editions,  as  follows  :  Paris, 
1495;  Venice,  1505,  fol.;  Vienna,  1515  (p.  117);  and  a  com- 
mentary by  Vittori  appeared  at  Bologna  in  i  506.  Some  of  these 
editions  contained  two  or  three  of  his  works  in  one  volume,  and 
it  is  probable  that  his  arithmetic  and  his  treatise  on  proportion 
appeared  in  other  editions  than  those  mentioned. 

Bradwardin  was  one  of  the  earliest  English  mathematicians  after 
Baeda  and  Alcuin.  His  arithmetic  is  of  the  Boethian  type,  relating  to 
the  theory  of  numbers.  He  gives  much  attention  to  the  ancient  theory 
of  ratios  and  to  figurate  numbers. 


62  RARA   ARITHMETICA 

BOETHIUS,    JORDANUS  NEMORARIUS,  and  FABER 
STAPULENSIS.     Ed.  pr.  1496.  Paris,  1496. 

For  the  biography  of  Boethius  see  p.  25. 

JoRDANUS  Nemorarius  (Jordanus  de  Saxonia)  was  born  at  Borgent- 
reich,  in  the  diocese  of  Paderborn,  and  died  in  1236.  He  studied  at  Paris 
and  was  the  greatest  mathematician  of  his  time  save  Leonardo  Fibonacci 
of  Pisa. 

Jacobus  Faber  Stapulensis  (Jacques  le  Fevre  d'Estaples)  was  born 
at  Estaples,  near  Amiens,  in  1455,  ^^'-^  died  at  Nerac  in  1536.  He  was  a 
priest,  vicar  of  the  bishop  of  Meaux,  lecturer  on  philosophy  at  the  College 
Lemoine  in  Paris,  and  tutor  to  Charles,  son  of  Franfois  I.  He  wrote  on 
philosophy,  theology,  and  mathematics. 

Title.    See  Fig.  37. 

Description.  Fol.,  20  X  29  cm.,  the  text  being  13.9  X  27.2  cm. 
72  ff.  unnumb.,  60-63  11.     Paris,  1496. 

Editions.  The  arithmetic  of  Jordanus  went  through  various 
editions  as  follows:  Paris,  1496,  fol.  (here  described);  ib.,  1503, 
fol.;  ib.,  1 507,  fol.  (p.  65);  ib.,  15 10,  fol.;  ib.,  15 14  (p.  65).  Rogg 
speaks  of  an  edition  of  1480,  but  I  do  not  know  of  it.  Jordanus 
also  wrote  an  '  Algorithmus  demonstratus,'  published  anony- 
mously at  Niirnberg  in  1534,  4°.  De  Morgan,  following  Scho- 
nerus,  attributed  it  to  Regiomontanus,  but  the  evidence  shows 
that  he  only  revised  it  and  it  may  be  due  to  Jordanus.  His  'De 
Ponderibus,'  edited  by  Apianus,  was  published  at  Niirnberg  in 
1533,  and  at  Venice  in  1565.  An  interest  attaches  to  the  1496 
edition  in  that  it  is  the  first  printed  work  with  which  a  Scotch- 
man's name  is  connected,  the  printer  being  David  Lauxius  of 
Edinburgh,  then  working  in  Paris. 

The  greater  part  of  this  volume  is  devoted  to  the  ten  books  on  arith- 
metic by  Jordanus  Nemorarius,  with  the  commentary  of  Jacobus  Faber 
Stapulensis.  The  work  of  Jordanus  is  similar  to  that  of  Boethius,  and  is 
concerned  only  with  theory  of  numbers.  In  particular,  the  Greek  theory 
of  ratios,  as  elaborated  during  the  Middle  Ages,  is  extensively  treated. 

The  second  part  consists  of  the  work  of  Jacobus  Faber  Stapulensis  on 
music,  in  four  books. 

The  third  part  is  the  Epitome  of  the  Arithmetic  of  Boethius : 
'  <LJacobi  Fabri  Stapulenfis  Epitome  in  duos  libros  Arithmeticos//diui 
Seuerini  Boetij  ad  Magnificum  diium  Joannem  Stephanum  //  Ferrerium 
Epifcopum  Verfellenfem.' 


PRINTED    BOOKS  63 

The  fourth  part,  consisting  of  four  and  a  half  pages,  is  a  dcsc  ription 
of  the  arithmetical  game  of  Rithmimachia,  possibly  by  Shirewode  (John 
Shirwood,  Bishop  of  Durham,  who  died  in  1494),  but  usually  ascribed 
to  Faber  Stapulensis.    An  edition  appeared  at  Erfurt  in  1577,  4°. 

5nbocopcrccontcnta. 
Sritbmcrica  occcm  libn's  ocmonflrdtt 
^fcufica  lib:i0  Dcmonftrata  quattuot 
/epitome  I  lJb:o0  aritbmcncos  Dim  Scucrini  :eoctii 
lR.tbmimacbicluouff4«PW0n«n»'^«''O9appclla5 


i6.«5ontcrius  iCabiloncnfietfrt 
lauoe  arirbmcriccs  z  Jihuficco* 

Ccmpozc  iam  muUo  Doctc  latucrc  fo:o:c9: 

(gluao  rctinct  comitca  flaua  mincrua  fuas. 
TBuc  pbaoc  icrrae  port  icmpon  mulia  rcuifunt* 

i6rm  quocp  ante  alias  «5alUca  terra  placet. 
Ibi0  oUm  Celebris  fuit  omnio  Scaica  tcllus. 

•fl!>Ttba0oza  patriam  Diffugicnie  famon. 
Ibcllaoa  nunc  Unquimt:  ct  ooctas  pallaois  rrbc0* 

£>cquaniof(p  petunt/parbidofcp  lares. 
It?cc  venit  omnimooa  numero:um  dncta  caterua* 

atcp  oocct  numcris  quioquio  in  o:bc  fitum  eft. 
aitCTaoulcifono  cantu/ftoibufcp  canozis 

^oomuilTe  viros  traoitur  atcp  fcras: 
£lue  fua  ipicrijs  tenet  vnica  nomma  muils 

^  nicbil  bac  murts  gratius  elTe  folcu 
Zttmcn  artiftcem  ftapule  mifcre  manne: 

^ui  Icta  boc  ftuDijs  frontc  oicaretopus. 
Iboc  folum  ftuDium  atcp  bee  ilU  cura:iuuare/ 

^rritus  ct  nc  fit/oifpercafcp  labo:. 
Fir;.  37.    Title  page   of  the   1496  hoethius 

Of/ier  7i'orks  0/1496.  Albert  of  Saxony,  p.  9,  c.  1478  ;  Bradwardin, 
p.  61,  1495  ;  Z.  Lillius,  '  I)e  origine  et  laudibus  scientiarum,'  Florence, 
4°  (one  page  on  arithmetic).    There  was  also  published  at  Pans,  s.  a. 


64  RARA   ARITHMETICA 

(c.  1496),  an  anonymous  treatise  entitled  '  De  arte  numeradi  copediu 
putile  icipit  feliciter.  (Q)uoniam  rogatus  a  plurib5  compedium  artis 
numerandi  ac  breue  tractalutu  .  .  .' 

JForl's  0/1497.    Boethius,  p.  27,  1488  (the  colophon  of  the  1497 


E 

^ 

n 

0 

1  0^1 

n 

\ -bp  / 

□ 

m 

d 

^ 

qi 

^  001  ^ 

\o6   / 

□ 

© 

0 

s 

y 

R^ 

a 

q 

S 

q 

i 

\ 

B 

@ 

S 

@ 

A 

S 

@ 

H 

® 

P] 

A 

w 

♦ 

B^3fi 

A 

m 

A 

^ 

A' 

^ 

^ 

± 

M 

4^ 

Fii;.  38.    RrrHMiMACHiA,   1496  boethius 

edition  has  the  incorrect  date  Mcccclxxxvii)  ;   Suiseth,  1497  and  1498, 
p,  10,  c.  1480. 

IVorks  of  I4g8.  Anianus,  p.  32,  1488;  Bradwardin,  p.  61,  1495  ; 
Chiarini,  p.  11,  1481  ;  Anonymous,  'Enchiridion  sive  tractatus  de 
numeris  integris,  fractis,'  etc.,  4°  (doubtless  the  work  published  at 
Deventer  in  1499,  p.  67). 


PRINTED    HOOKS  65 

BOETIIIUS,  JORDANUS   NEMORARIUS,   and    FAHKR 
STAPULENSIS.     Ed.  pr.  1496.  Paris,  1507. 

See  p.  62. 

Title.  'In  hoc  opcrc  contcnta.// Arithmctica  dec-cm  libris 
demonftrata // Mufica  libris  demonftrata  quattuor // Epitome  1 
libros  arithmeticos  diui  Seuerini  Boetij  //  Rithmimachie  ludus 
q  I  pugna  nQerop  appellat.'     (F.  i,  r.) 

ColopJion.  '  Cimprcffum  Parifij  in  ofificina  Ilenrici  ftejjhani 
e  regione  Scheie  decretorum  rita.//Anno  Chrifti  fiderum  con- 
ditoris  1507.     Decimo  die  Nouembris.'    (F.  78,  r.) 

Description.  8°,  20x26.6  cm.,  the  te.xt  being  18.1x21.5 
cm.     78  ff.  unnumb.  +  i  blank  =  79  ff.,  60  11.     Paris,  1507. 

Editions.  See  pp.  27,  62,  for  the  editions  of  Boethius  and 
Jordanus. 

Like  most  of  the  works  from  the  press  of  Stephanus,  this  book  is 
beautifully  printed.  It  contains  the  '  Elementa  Arithmetica  '  of  Jorda- 
nus, with  the  demonstrations  of  Faber  Stapulensis,  the  Epitome  of  Boe- 
thius by  Faber  Stapulensis,  the  '  Rithmimachia,'  the  commentary  on 
Sacrobosco's  astronomy  by  Faber  Stapulensis,  and  the  first  four  books 
of  Euclid  '  a  Boetio  in  latinum  tranflate.' 

BOETHIUS,  JORDANUS   NEMORARIUS,  and   FABER 
STAPULENSIS.     Ed.  pr.  1496.  Paris,  1514. 

See  p.  62. 

Title.  '  In  hoc  opere  contenta //Arithmetica  decem  libris// 
demonftrata.// Mufica  libris  demoftrata // quatuor.// Epitome 
in  libros  Arith-//meticos  diui  Seuerini // Boetij.// Rithmima- 
chie ludus  qui // et  pugna  numerorij  ap-//pellatur.'//  (Sur- 
rounded by  an  elaborate  woodcut  with  the  following  wording : 
'  Haecsecvndariaes//tetcastigat//issimacxofificina//aemissio'  // ) 
'  CHec  fecundaria  fuperioru  operum  seditio  //  venalis  habetur 
Parifijs://in  olficina  Henrici  Stephani  e  regione  fchole  Decre- 
torum.'   (F".  I,  r.) 

Colophon.  '  CHas  duas  Quadriuij  partes  et  artium  liberalium 
precipuas  atq5  duces  cu  quibufdam  ammini-//cularijs  adiectis: 
curauit    e.x    fecunda    recognitione    vna    fornuilis    eniedatiffime 


66  RARA   ARITHMETICA 

mandari  ad  ftudiorum // vtilitatem  Henricus  Stephanus  fuo 
grauiffimo  labore  et  fumptu  Parhifijs  Anno  falutis  domini :  // 
qui  omnia  in  numero  atq5  harmonia  formauit  15  14.  abfolutumqs 
reddidit  eode  anno:  die  feptima // Septembris/ fuum  laborem 
vbicunq5  valet  femper  ftudiofis  deuouens.'    (F.  71,  v.) 

Description.  Fol.,  19.8  X  28.4  cm.,  the  text  being  17.7  X  26.8 
cm.    71  ff.  unnumb.,  62  11. 

Editions.  See  p.  62.  This  edition  is  practically  identical  with 
that  of  1496.  It  is  the  second  edition  of  this  combination  of 
works  and  the  fourth  of  Faber's  Epitome. 

MARTIANUS   MINEUS   FELIX   CAPELLA. 

Ed.  pr.  1499.  Vincenza,  1499. 

Flourished  c.   475.     He  was  probably  born  at   Carthage,  and  he   lived 
at  Rome. 

Title.    See  Fig.  39. 

Colophon.  'Martini  Capellae  Liber  finit:  Impreffus  Vincentiae 
Anno  Salutis  //  M.ccccxcix.  xvii.  Kalendas  lanuarias  per  Henri- 
cum  de  Sancto  //  Vrfo  Cum  gratia  &  priuilegio  decem  annorum: 
ne  imprima-//tur  neqs  cum  Commentatiis:  neqs  fine:  &  caetera: 
quai  in  ipfo  pri//uilegio  continentur.  Laus  Deo  &  beatae 
Virgini.'     (F.  123,  v.) 

Description.  Fol.,  20.5  X  30.3  cm.,  the  text  being  12. i  X  22.3 
cm.     124  ff.  unnumb.,  37  11.     Vincenza,  1499. 

Editions.  Vincenza,  1499,  fol.  (here  described);  Modena,  1 500, 
fol.  (p.  67);  Vienna,  1516,  fol.;  Basel,  1532,  fol.;  Leyden,  1539; 
Basel,  1 577,  fol.  (p.  68) ;  Leyden,  1 592,  8°  (p.  68) ;  ib.,  1 599,  8°  ;  ^ 
and  later.     An  Italian  translation  was  published  at  Mantua  in 
1578.    For  bibliography,  see  Boncompagni's  Bulletino,  XV,  506. 

This  work  is  a  medley  of  prose  and  verse,  and  forms  a  kind  of  ency- 
clopedia of  the  arts  and  sciences  as  known  for  about  a  thousand  years. 
It  was  highly  esteemed  in  the  Middle  Ages  as  a  textbook.  The  seventh 
book  is  on  the  Greek  theory  of  arithmetic.  It  treats  of  the  various 
classes  of  numbers,  such  as  plane  and  solid,  and  mentions  the  supposed 
mysteries  of  the  smaller  numbers,  the  monad  suggesting  one  God,  the 
dyad  good  and  evil,  the  triad  the  Trinity,  and  so  on. 


PRINTED    BOOKS  67 

Other  7vorks  of  I4gg.  Boethius,  p.  27,  [488;  Suisclh,  p.  10,  c. 
1480  ;  Anonymous,  '  Enchiridion  Algorismi  sive  tractatus  dc  numeris 
integris,'  Deventer,  4°  (p.  64,  1498)  ;  (ieorge  of  Hungary,  '  Arithmetica 
summa  tripartita,'  s.  1.,  reprinted  at  Budapesth  in  1894. 

Opn9 

i   ©artianiCapdlede^upttj^ 

^  Dedukcclca* 
6-  x^cgcoin^tri  ♦ 

j  DC  mafia  librifcprcm. 

Fig.  39.   TrrLE  page  of  the   1499  ^'ai'klla 

MARTIANUS   MINEUS   FELIX    CAPELLA. 

Ed.  pr.  1499.  Modcna,  1500. 

See  p.  66. 

Title.  'Opvs.//  Martiani  Capellae  //de  nvptiis  phi//lologi3e  et 
//mercvrii//liberi//dvo.//De  gramatica.  Liber.  Tertius.//De  di- 
alectica.  Liber.  Quartus.//DeRhetorica.  Liber.  Quintus.//Dege- 
ometria.  Liber.  Sextus.//De  Arithmetica.  Liber.  Septimus.//De 
aftronomia.  Liber.  Octauus.//De  mufica.  Liber.  Nonus.'  (F.  i,  r.) 

Colophon.  '  Martiani  Capellse  Liber  finit.  Impreffus  Mutinae. 
Anno  Salutis.  M.//CCCCC.  Die  .XV.  Menfis  Maii.  Per  Dionyfiu. 
Berthocum.'     (F.   100,  r.) 

Description.  Fol.,  20  X  29  cm.,  the  te.xt  being  13.7  X  24  cm. 
On  f.  69,  v.,  the  part  on  arithmetic  begins,  and  occupies  10  ff. 
100  ff.  in  the  entire  book,  unnumb.,  42  11.     Modena,  1500. 


68  RARA   ARITHMETICA 

MARTIANUS   MINEUS   FELIX   CAPELLA. 

Ed.  pr.  1499.  Basel,  1577. 

See  p.  66. 

Title.  '  Isidori  //  Hispalensis  //  Episcopi  //  Originum  libri 
viginti  //  ex  antiquitate  eruti.//  Et  //  Martiani  Capellae  //  De 
nuptijs  Philologiae  &  Mercurij  //  Libri  nouem.// Vterque,  praeter 
Fulgentium  &  Veteres  Grammaticos,  va-/Aijs  lectionibus  & 
fcholijs  illuftratus  //  Opera  atq  Industria  //  Bonaventvrae  Vvlca- 
nii  Brvgensis.//  Cum  gratia  &  priuilegio  Caefareae  Maieftatis. 
//Basilese,//per  Petrvm  Pernam.'    (F.  i,  r.) 

Description.  Fol.,  20.5  X  31  cm.,  printed  in  double  columns, 
each  being  8.2  X  24.4  cm.,  7  ff.  +  240  columns  +  index  +550 
columns.  The  part  on  arithmetic  in  the  work  of  Capella  begins 
in  column  155  and  covers  12  pp.,  or  as  here  numbered  24  col- 
umns, 60  11.     Basel,  1577. 

Editions.    See  p.  66. 

This  edition  includes  the  works  of  both  Isidorus  (p.  8)  and  Capella, 
and  is  an  excellent  specimen  of  printing.  One  interesting  feature  of 
Capella' s  work  is  the  evidence  that  it  gives  of  the  use  of  the  abacus  in 
the  fifth  century.  We  are  still  quite  uncertain  as  to  the  history  of  this 
method  of  calculating  in  the  centuries  following  Capella. 

MARTIANUS   MINEUS  FELIX    CAPELLA. 

Ed.  pr.  1499.  Leyden,  1592. 

See  p.  66. 

Title.  'M.  Capella.// Martiani //Minei  Capella; // Carthagi- 
nensis  //  de  Nvptiis  Philolo-//giae,  &  feptem  artibus  //  Libe- 
ralibus  //  Libri  Novem  //  optime  castigati.//  Lvgdvni,//  Apud 
Bartholomaeum  Vincentium.// 1592.'    (P.  i.) 

Colophon.  '  Lugduni,//  Excvdebat  //  Stephanus  Seruain.// 
1592.'    (P.  416.) 

Description.  8°,  10  x  15.7  cm.,  the  text  being  7.1  x  12.8 
cm.  4  pp.  blank  +  18  unnumb.  +  396  numb.  =418  pp.,  30  11. 
Leyden,  1592. 

Editions.    See  j).  66. 


PRINTED    BOOKS  69 

totuptilrinfergtiitSiF  i^toule 
Umrffptf  firamonfitregrts  Cona 
!ft^^rtmBtffpIi5itontm  i^me 

(icntinfcolis  ^ornbergen.flntbmcirico2fi  wed. 
^nflozenriffimo  fltudio  aipcjcnfinuecdit^THo 
mmnd  btrem  cruditis  ^  ^crcatoixhne  \tiliB  % 
vmxxat  incipientibo^. 

^urea  fuccinctepateattibirfgiila  tsetrC 
^rartgere  quo  raleas  qaem  rmfmta1"vafer 

21  fccMQ  Dictas  quo  p^ofTis  pjedcrej nczmas 
IDuttid  viUfcant  non  tibi  oona  libn' 

Tr>i|8  fiurnberga  nitet  uumerat?  di  mfignie  ab  arte  ^ 
IDoic  artt  mulrom  contvMx  tlla  bont 


Fig.  40.    Title  page  ok  light 


70  RARA   ARITHMETICA 

BALTHASAR    LIGHT. 

Ed.  pr.  1500.  Leipzig,  s.  a.  (1500). 

A  German  Rechenmeister  of  c.  1500. 

Title.    See  Fig.  40. 

Colophon.   ' I mpreff um  Lipczk  per  Melchiar  Letter.'   (F.  I5,v.) 

Dcscriptioji.  4°,  14  X  19.9  cm.,  the  text  being  9.3  X  16.2  cm. 
15  ff.  unmmib.,  22-37  11.     Leipzig,  s.  a.  (1500). 

Editions.  Leipzig,  s.  a.  (1500),  here  described;  ib.,  1509; 
1 5 13;  Leipzig,  15 1 5,  4°,  which  may  be  the  '  Algorithmus 
linealis,'  s.  a.  and  1505,  by  Letter,  referred  to  by  De  Morgan 
(p.  loi).  There  was  also  an  'Algorithmus  Hnealis,  Impressum 
Lipzik  per  melchiorem  Lotter  Anno  xc,'  probably  printed  in 
1490,  Lotter  having  printed  in  Leipzig  from  1490  to  15 12. 
(See  contra,  Abhandlungen,  V,  154,  n.,  152,  n.,  and  cf.  Widman, 
c.  1490,  p.  44.)  On  f.  I,  v.,  the  dedicatory  epistle  closes  with 
the  words  'Vale  ex  noftra  academia  Lyptzeh  Anno  1500,' 
which  throws  much  doubt  on  the  conjecture  that  Licht's  work 
appeared  earlier. 

This  is  a  brief  treatise  on  the  line  abacus,  one  of  the  earliest  of  the 
type  represented  also  by  Huswirt  (see  p.  73). 

LEONARDUS   FORTIUS. 

Ed.  pr.  c.  1500.  S.  1.  a.  (Venice  .?,  c.  1500). 

A  Venetian  jurist  of  the  fifteenth  century. 

Title.  '  Leonardi  //  de  Portis  ivrisconsvlti  Vi//centini  de 
sestertio  pe//cvniis  ponderibvs  et  //  mensvris  antiqvis  //  libri 
dvo.'     (F.  I,  r.    See  Fig.  41.) 

Description.  4°,  14.4  X  19.8  cm.,  the  text  being  10  X  14.8  cm. 
37  ff.  unnumb.,  30  11.    Venice  (?),  c.  1500. 

Editions.  S.  1.  a.  (Venice  .?,  c.  1500,  here  described); 
Florence,  15 14  (.?)  ;   Basel,  1520,  4°;  ib.,  1530,  8°. 

A  work  on  ancient  measures,  using  the  Roman  numerals  throughout, 
except  in  the  index.  Such  treatises  are  of  value  in  studying  the  history 
of  arithmetic,  but  are  not,  in  general,  included  in  the  bibliographical 
lists  of  this  work. 


PRINTED    BOOKS  71 

Other  works  of  c.  i^oo.  Anianus,  c.  1500.  p.  32,  1488;  Boelhius, 
c.  1500,  p.  27,  1488  ;  Bradwardin,  c.  1500,  p.  61,  1495  ;  Capella,  1500, 
p.  66,  1499;  Peurbach,  c.  1500,  p.  53,  1492.  Widman,  1500,  p.  37, 
1489.  There  also  appeared  about  this  time,  s.  1.  a.,  an  anonymous 
'  Algorithmus  minutiarum  vulgarium,'  blackletter,  4°  (Libri,  1861  cat., 
483),  and  an  anonymous  '  Ars  numerandi,'  5  ff.,  4°,  a  title  given  to 
several  books  of  this  ])eriod  (see  p.  23),  including  '  De  arte  numerandi 
siue  arismetice  (perfectionis)  summa  (luadripartita '  {Abhandlungen^  1, 
24  ;  Brunet,  Man.,  6(1),  45^)- 

LEONARDI 

DE  PORTISIVRISCONSVLTI  VI 

CENTINI  DE  SESTERTIO  PE 

CVNIIS  PONDERIBVS  ET 

MENSVRIS  ANTlCi^VIS 

LIBRI  OVO, 

Fig.  41.    Title  pa(;e  of  leonardus  fortius 
GEORGIUS  VALLA.     Ed.  pr.  1501.  Venice,  1501. 

Born  at  Piacenza  in  1430;   died  at  Venice  in  1499.    He  was  a  physician 
and  philologian. 

Title.  '  Georgii  Vallae  Placentini  viri  cla-//riss.  de  expetendis, 
et  fvgiendis  //  rebvs  opvs,  in  qvo  haec  //  continentvr.//  De  arith- 
metica  libri  .ill.  ubi  quaedam  a  Boetio  praetermiffa  tractantur.// 
De  Mufica  libri  .v.  fed  primo  de  inuentione,  &  commodiatate 
eius.//De  Geometria  libri  .vi.  in  quibus  elementorum  Eiiclidis 
difficultates  omnes  fere //exponuntm",  ubi  etifi  de  Mechanicis 
fpiritalibus,  Catoptricis,  ac  Opticis,  deq;  //  quadrato  circuli  habe- 
tur  tractatus.//De  tota  Aftrologia  libri  .iiii.  in  qua  fabrica.ufusq; 
aftrolabi  exaratur,  &  quae  fi-Z/gnorum  in  exhibendis  medica- 
minibus  fit  habenda  obferuatio.// De  Phyfiologia  libri  .iiii.  ubi  & 


72  RARA   ARITHMETICA 

Metaphyfices  qda  lectu  q  dignifs.  utiliffimaq;.// De  Medicina 
libri  .vii.  ubi  de  fimplicium  natura  per  ordinem  litterarum.// Pro- 
blematum  liber  unus.//  De  Grammatica  libri  .iii.//  De  Dialectica 
libri  .iii.//  De  Poetica  liber  unus.//  De  Rhethorica  libri  .ii.//  De 
Morali  Philofophia  liber  unus.//  De  Oeconomia,  fiue  adminiftra- 
tione  domus  libri  .iii.  in  quibus  de  Architectu-//ra,  req;  ruftica  eft 
locus.//  Politicon  unicum  uolumen,  ubi  de  iure  ciuili,  ac  pontificio 
primum,  Mox  de  le//gibus  in  uniuerfum,  Inde  de  re  militari 
agitur.//De  Corporis  commodis,  &  incommodis  libri  .iii.  quorum 
primus  totus  de  ani-//ma,  Secudus  de  corpore,  Tertius  uero  de 
urinis  ex  Hippocrate,  ac  Paulo  Degi-//neta,  deq;  Galeni  quaeftioni- 
bus  in  Hippocratem.//  De  Rebus  externis  liber  unus,  ac  ultimus, 
ubi  de  Gloria,  Amplitudine,  &  caete-//ris  huiufmodi.//  Haec  fum- 
matim,  fed  infunt,  &  alia  plurima,quae  legedo  licet  cognofcere.' 
(F.   i,r.) 

Colophon.  '  Venetiis,  in  aedibus  Aldi  Roma-//ni  impensa,  ac 
studio  Ioan-//nis  Petri  Valise  filii  pi-//entiss.  mense  Decern-// 
bri.    M.D.I.'    (Not  in  this  copy.) 

Description.  FoL,  28.8  X  43.7  cm.,  the  text  being  18.8  X  32.7 
cm.     3  ff.  blank  +  308  unnumb.  =  311  ff.,  55  11.     Venice,  1501. 

Editions.    There  was  no  other  edition. 

The  first  book  consists  of  23  brief  chapters  on  the  general  value  and 
nature  of  mathematics  (18  pp.);  the  second  book,  of  18  chapters  on 
the  Greek  classification  of  numbers  (17  pp.)  ;  the  third,  of  20  chapters 
on  figurate  numbers,  proportions,  and  the  fancied  properties  of  each 
number  of  the  first  decade  (27  pp.);  and  the  fourth,  of  13  chapters 
on  the  operations  (13  pp.).  There  is  nothing  that  is  noteworthy  in 
the  treatment.  Works  on  the  value  of  mathematics  were  quite  common 
at  this  time,  while  all  university  treatises  on  arithmetic  were  devoted 
chiefly  to  the  Greek  theory.  The  thirteen  pages  devoted  to  the  opera- 
tions were  a  rather  generous  allowance  for  the  time,  especially  as  each 
page  has  as  much  matter  as  six  or  eight  pages  of  an  ordinary  octavo 
arithmetic  of  that  period. 

Valla  also  wrote  a  treatise  on  the  astrolabe,  '  Insignis  philosophi 
Nicephori  Astrolabii  expositio '  (Paris,  1554),  and  published  an  edition 
of  Euclid  (Venice,  1492).  His  collected  commentaries,  but  without  the 
arithmetic  and  other  original  works,  appeared  in  Venice  in  1498. 


I^RINIIU)    HOOKS 

ARNALDO    DK   VILLA    NOVA. 
Kd.  pr.   I  501. 


73 


Venice,  I  501, 


Arnaui.I)  1)K  Vii.LENEUVE,  Arnali)  Baciiuone.  IJom  ill  12^8,  at  Villa 
Nova  (Catalonia),  or  possibly  Villeneuve,  near  Montpellier ;  died  in  IJ14, 
shipwrecked  on  the  Mediterranean.  He  is  known  princii)ally  for  twenty 
works  on  alchemy.  He  lectured  on  philosophy  and  medicine  at  Barcelona 
and  Paris,  and  was  later  a  celebrated  physician. 

Computus  £cclciyfticu6rSftrono^ 

mictt6-£ditU6  a  ZI^3giftro  Br* 

naldo  oe  villa  *Ploua  "^o.* 

ttitcr  5mp:eflUm, 


jCttm^:ati3  -6t  nfynuilegio. 


Fif;.  42.    Title  page  of  akn.aluo  de  \ii.l. 


74  RARA   ARITHMETICA 

Title.  '  Computus  Ecclefiafticus  3  Aftrono-//micus  Editus  a 
Magiftro  Ar-//naldo  de  villa  Noua  No-//uiter  Impreffum.// 
Cum  Gratia  Et  Priuilegio.'     (F.   i,  r.    See  Fig.  42.) 

Colophon.  'Cimpreffum  Venetijs  per  Bernardinu  Venetu 
deVitalibus.//Anno  Dni.M.CCCCC.J.Die  .xvij.MeH.Februarij.' 
(F.  II,  V.) 

Description.  4°,  14.4  X  20.9  cm.,  the  text  being  10.5  X  16.6 
cm.     II  ff.  unmimb.,  37-39  ff.     Venice,  1501. 

Editions.    There  was  no  other  edition. 

This  is  a  good  example  of  the  works  on  the  ecclesiastical  calendar 
in  use  in  the  Middle  Ages.  It  employs  only  the  Roman  numerals  and 
gives  no  treatment  of  computation.  In  spite  of  the  words  '  nouiter 
impressum,'  I  know  of  no  earlier  edition,  and  indeed  these  words  were 
not  infrequently  used  when  a  book  was  first  printed. 

JOHANN    HUSWIRT,  Sanensis. 

Ed.  pr.  1 50 1.  Cologne,  1501. 

A  German  arithmetician  of  c.  1500.  The  name  Sanensis  suggests  his 
birthplace  as  vSayn  in  the  Westerwald,  and  the  problems  relate  to  places  in 
that  vicinity.    Nothing  is  known  of  his  life. 

Title.    See  Fig.  43. 

Colophon.  '  Enchiridion  algorifmi  fagaci  cura  ftudioq5  p 
Johane  hufwirt  sanefez // elaboratus.  caracteri  p°  ;3miffus  Colo- 
nie  In  officina  felicis  memorie  ho//nefti  viri  Henrici  Quentell. 
Anno  repatoris  humane  feruitut5  Mccccci.'     (F.  20,  r.) 

Description.  4°,  14.3  X  20.6  cm.,  the  text  being  9  X  15.4  cm. 
20  ff.  unnumb.,  25-47  11.     Cologne,  1501. 

Editions.  Cologne,  1501,  4°  (here  described);  ib.,  1503;  ib., 
1504,  4°  (p.  77);  1507;  1554;  and  a  French  edition  (Chasles). 
It  was  published  with  historical  notes  by  Professor  Wildermuth, 
at  Tiibingen,  in  1865.  Mr.  Plimpton's  copy  has  the  bookplates 
of  Dr.  Kloss  and  Chasles. 

This  is  the  earliest  treatise  on  algorism  printed  at  Cologne.  It  is 
divided  into  four  '  tractati,'  and  includes  the  fundamental  operations 
through  evolution  (' Tractatus  Primus'),  a  brief  treatment  of  abacus  or 
line  reckoning  ('  Cl'ractatus  Secundus  de  proiectilibus  '),  common  frac- 
tions ('Tractatus  Tertius'),  rule  of  three,  partnership,  and  over  twenty 


PRINTKD    BOOKS  75 

miscellaneous  rules  ('Tractatus  fjuartus  de  regulis  mercalo^ '  eU.). 
In  the  algoristic  treatment  of  integers  Huswirt  places  '  duplatio 
(doubling)  after  multiplication,  and  '  mediatio  '  (halving)  after  division  ; 


^ 


fine  figurani  (moit^zlo  ^)  dc/ 
Unoncecomotk  traai^ 

tusfitmncTum 
mcneccf/ 
fan/ 

■19 


tnuiJc  ticTatt'esflmgua  compcTccRirentenn 
Ncc  nimiu.  rabidis  gamilus  cfto  labris* 

Autpctetartareascfupcnsincognittss)  vmbras 
Et  phlsgetonreos  labereadulq^  lacus 

Atcp  illi'cpotiuslitcs*  &iurgia  mifcc 
^cviiuspac^mconcubiorclinc 

Fio.  43.    Title  pace  of  the   1501   huswirt 

but  vi'hen  he  is  dealing  with  counters  and  with  fractions  he  places  them 
before  multiplication,  because  they  are  needed  there  in  abacus  calculating. 
It  is  interesting  to  see  how  these  chapters  on  doubling  and  halving,  of 
which  we  have  traces  in  ancient  Egypt,  persisted  throughout  the  Middle 
Ages  and  well  into  the  sixteenth  century. 


76 


RARA   ARITHMETICA 


As  in  several  other  works  of  this  period,  there  is  evidence  of  the 
difficulty  of  finding  a  generally  acceptable  name  for  the  character  0, 
a  difficulty  not  yet  removed  in  the  English  language.  Husvvirt  gives 
four  names  to  this  tenth  character  :  '  Decimo  X/o  theca.  circul?  cifra. 
fiue  figura  nihili  appellat'.' 

There  is  also  noticeable  in  this  work  a  tendency  which  is  seen  in 
other  arithmetics  of  the  time,  to  name  a  group  of  problems  after  some 
well-known  type.  For  example,  Huswirt's  sixth  rule  is  that  of  the  flee- 
ing hare  ('  Regula  Sexta  de 


lepore  fugiente,'  f.  i6,  r.), 
although  it  has  nothing  to 
do  with  the  hound  and  hare, 
but  relates  to  a  traveler  going 
from  Cologne  to  Rome,  and 
followed  five  days  later  by 
another  traveler  who  in 
due  time  overtakes  him. 
('  CAmbulat  qdam  de  Colo- 
nia  verfus  roma  et  ambulat 
qtidie  9  miliaria,  alius  aiit 
pfequit'  ipfum  poft  5  dies,' 
etc.) 

Other  works  of  i^Oi. 
Anianus,  p.  32,  1488;  Sa- 
crobosco,  p.  32,  1488  ;  Boe- 
thius,  p.  27,  1488;  Borghi, 
p.  16,  1484  ;  H.Torrentini, 
'  Elucidarius  carminum  et 
hystoriarum,'  Deventer,  4° 
(with  a  chapter  on  arith- 
metic), with  subsequent  edi- 
tions as  follows:  ib.,  1503, 
4°;    Strasburg,    1505,    4°; 


m 


f t)e  multiplicfltione   Capl'm  quartiiitt 
Uciplicario  cd  numeri  p2ocrcatio.p:opo:tion3bi 

Uircr  fe  tjakntis  ad  multiplicandu  fic  ut  mulnpliMiic  ad  vni> 
Frarcmfeijahrwjctnipli  gratia5ad  4  mulnphcarc  eft  mime 
J  rum  |r  pjocrcarc.  que  lie  multiplicando  videlicet  4  pjo^ 
p;ocionanrur  qucttiadmodutnmiilnplicansrciliccr }  vnirati  concCponJ 
Dct.quia  vrraif  eft  pjopoino  mpla.^rcm  multiplicacio  pjcrcqmnt  <r  q» 
hnc  mulriplicarioncm  Pigitojum  mrcr  k  fciar.Cuius  talis  oamr  rc'Tuli 
ScribjnrurMgirifiibalrcme.etcuiiinibttDiffcrcntiaaOenano  verfus  PC 
%xxm\  ponae.quae  mrcrfcmuInplifa.etpjoductuminfcnuefcnh-DnnJ 
e«  Oifferenriam  vniusaDigitoaltcrius  fufafra^eefprwipjoducto  p«iW 
ponc.rtpioucnictfumma.vtparctinfigurj  £j[cmplum.(criceS 

bisimo    8  s    Differcnta  quotliintJnuiripiica 

Ciffcrciniaseomaoc 

©ifiifua    «s  4    CifTcroiria  ti3no.fciUc}4«i"» 

48  fcrf«rcrunf8,que 

ftntr-S^cindetiffercnriam  vniueaCigitoalfm'us  fubcraljc-etrrficfuni 
Kioii  pwducto  poftponc.vrparet  m  figura.  ^'S.H  rcgnw  ^<:  niuinP"l[ 
canoneoiioiumnumcrojum infra  10  quojum  quilittt  DuJbus  ngiiri* 
fcnprus  crt.pjopofitis  itaii;  Duobue  numtnsppmasinfcnoiiB  cumpma 
fiipcriona  muJtiplica^t  piocrcabirur  numerus  vna  vdDuabjfxguns  icre 
Itndu0.rivna.fmbarur.fiouab5.p:itnamljarumfcnh:.r«midamlcruan7 
CO  in  mtntf.Demde  trerumcafdcmadfe  addae.ctpzodurtopwcJcn/ 
temfiguramfn  menre  refcruaram  adiUnge-er  pzoueracf  "'"""^f  Pli,^ 
bus  figurts  xA vna fcntmdueSi vna.fcnbatur.et o"'"'^'^  ^f  ",f /'* 
ria!umnumcr«umrm  vn,casatc,p,Octet.quam  ponponcp:io.^b^^^^^^^^ 
cnr  fumma.Siautcmcuab.pnmam  (?9rum fcTih.fecudamvnitJttapi* 
ftcnonto  figutre aoipunde addcquae  ^'■'^'^^^^^'^^^l^^^^.  ,„-  a 
|£,»T,plmn  \    7  ^ntz8fmec8.s 

_  -j-^ —  rcnifldoimenre.iDe 

Mt  aide  pjimas  fig'uraa  ad  muitctn  «>'«"^;ten''Jqu3m'ldd^^^^^^ 

itomaMbuUUicpoftcaponwf 

Fk;.  44.   Complementary  multiplication, 
HuswiRT  (1501) 


Hagenau,     1507,    4   ;    ib., 

1510,  4°;  ib.,  1512,  4°;  Strasburg,  1514,  4°;  1515,  4°;  Strasburg, 
1518,  4°;  Paris,  1530,  8°;  ib.,  1535,  8°;  Cologne,  1536,  8°;  Paris, 
1550,  8°. 

Works  0/ IJ02.  Anianus,  p.  32,  1488;  Albert  of  Saxony,  p.  9,  c. 
1478  ;  Bradwardin,  p.  6,  1495  ;  Nicolo  Calvino,  a  work  on  arithmetic 
and  geometry,  Milan,  of  which  no  extant  copy  is  known  (see  Riccardi, 
part  I,  col.  213). 


PRINTED    HOOKS  77 

JOHANN    IIUSWIRT,  Sancnsis. 

Ed.  pr.   I  501.  Cologne,  1504? 

vSee  p.  74. 

Title.    The  same  as  in  the  echtion  of  1501  (see  p.  74). 

ColopJio)!.  The  last  folio,  with  the  colophon,  is  missing;  from 
this  copy. 

Description.  There  are  a  few  changes  in  type,  but  otherwise 
this  edition,  which  is  probably  that  of  1504,  is  line  for  line 
identical  with  that  of  1501. 

ANTON   BARTHOLOMEO   DI    PAXI. 

Ed.  pr.  1503.  Venice,  1503. 

Pasi.    a  Venetian  writer  of  the  fifteenth  and  sixteenth  centuries. 

Title.  '  Tariff  a  de  pexi  e  mesvre.//con  gratia  et  privilegio.' 
(F.  I,  r.)  '  Prohemio  del  prestantissimo  miser  Bartho-//lomeo  di 
Paxi  da  Venetia.'  (F.  i,  v.)  '  Qvi  comincia  la  vtilissima  opera 
chiama-//ta  taripha  laqval  tracta  de  ogni  sorte//de  pexi  e  mi- 
svre  conrispondenti  per  tvto  //  il  mondo  fata  e  composta  per  lo 
excelen//te  et  eximio  miser  Bartholomeo  di  Paxi  da //Venetia.' 
(F.  2,  r.     Fig.  45.) 

Colophon.  '  Stampado  in  uenefia  per  Albertin//da  lifona  uer- 
cellefe  regnante  lin-//clyto  principe  mifer  Leonardo  lo//reduno. 
Anno  domini.    1503.  A  di//26.  del  mefe  de  luio.    Finis.'  (F.  k  5.) 

Description.  4°,  15.5  X  21. i  cm.,  printed  in  double  columns, 
each  being  5.3  x  15.9  cm.     156  ff.  unnumb.,  33-38  11.     Venice, 

1503- 

Editions.  Venice,  1503,  4°  (here  described);  ib.,  c.  15  10,  4° 
(p.  79);  ib.,  1 52 1,  8°;  ib.,  1540,  8°  (p.  79);  ib.,  1557,  8°  (p.  80). 

The  book  is  not  a  textbook  on  arithmetic,  but  a  collection  of  infor- 
mation useful  to  merchants,  relating  to  the  measures  of  weight,  value, 
length,  etc.,  of  the  various  cities  and  countries  with  which  Venice  had 
trade  relations.  It  is  valuable  as  leading  to  an  understanding  of  the 
contemporary  arithmetics  of  Italy,  and  historians  could  find  much  useful 
information  as  to  the  prices  and  the  material  of  trade  by  examining 
this  and  similar  works.  An  inspection  of  Fig.  45  will  give  some  idea  of 
the  scope  of  Paxi's  Tariffa. 


7^ 


RARA   ARITHMETICA 


O'VICOMINCIALA  VTILISSIMA  OPERA  CHIAMA. 
TA  TARIPHA  LACLVAL  TRACTA  DE  OGNl  SORTE 
DE  PEXI  E  MISVRE  CONRlSPONDENTl  PER  TVTO 
IL  MONDO  FATA  E  COMPOSTA  PER  LO  EXCELEN 
TE  ET  EXIMIO  MISER  BARTHOLOMEO  Dl  PAXl  DA 
VENETJA. 

A VERemo  aduqj  prima  a  dechiarare  a 
uoftre  excelletie  tutte  lerobe  che  fe  ueda 
no  1  Venetia  a  pexo  groflb  &.  quelle  che 
fc  uedano  a  pexo  Ibtile  e  de  lordine  di  pe 
XI  de  li  ar2enri;6i  de  Ic  coditionc  di  pan' 
ni  de  lane  francefche  fatte  in  yenetia:8i 
etiam  dele  condirione  d^i  panni  de  feda: 
e  panni  doro:&:  del  ordine  del  uedere  de 
Ic  lpecie.'&  de  le  iue  tare;8{  del  ordine  di 
pcxi  dele  farine  e  bifchoti:  S^del  ordine 
de  le  miliirc  di  uini:&:  del  ordine  del  uen 
der  del  oio:&:  in  che  mode  e  pexo  fe  uen 
deno  iguadi:6C  del  ordine  di  frutti  che  fe  ucndeno  a  nome  6x  fter:  c 
che  pexo  hano  cadaun  fter:e  come  relponde  el  pexo  grofTo  con  tu 
ta  la  ltalia:6^  tuto  el  leuanre  6L  ponente;8i  etiam  come  ipexi  fubti^ 
li  e  pcxi  groffi  refpondeno  con  molte  terre  de  Italia ;  de  dalmaria  e 
de  Ieuante:&  come  refpondeno  le  mefure  di  panni  de  lana  con  tu^ 
ta  Italia:e  con  turo  el  leuante:8i  etiam  come  refpondeno  le  mefu^ 
re  di  panni  de  feda:e  panni  doro  c  darzcnto  con  tuta  ltalia:6^  co  tu 
to  el  leuante  &  ponentcSi  come  refpondeno  le  mefure  dingilterra 
zot  la  uirga  da  londra:e  de  la  taripha  dalixadL-ia;6«:  etiam  quella  de 
damafcho  8i  la  taripha  da  leppo.e  come  torna  la  fporca  dalixadria 
con  moire  terre  de  leuante  e  de  ponete  e  de  la  ltalia;6^  come  el  can 
ter  forfori  dalixandria  refponde  con  alchune  altre  terre  de  leuante 
de  ponente  e  de  ltalia;6d  etiam  come  refponde  el  canter  zeroui  con 
alchune  terre  del  leuante  del  ponente  e  de  ltalia:5^  come  refpode  cl 
cento  de  le  mene  con  alchune  terre  del  leuante  e  molte  de  Italia  SL 
del  ponente:8C  come  refpondeno  tuti  icantcra  de  leuante  e  del  po> 
nente  con  el  pexo  fubtile  da  Venetia:  &  come  relpode  e!  canter  da 
napoli  de  reame  con  molte  terre  del  leuante  del  ponete  e  con  mol" 
tc  de  Italia:^  come  refponde  el  canter  de  Conftatinopoli  con  moP 
t€  teirc  de  leuante  fic  etiam  de  ltalia:6^  come  refponde  el  canter  de 

Fig.  45.    The  beginning  of  the   1503  paxi        Z  ii 


PRINTED    HOOKS  79 

ANTON    BARTMOLOMEO   DI    PAXI. 

Ed.  pr.  1503.  S.  1.  a.  (Venice  ?,  c.   15  10). 

See  p.  77. 

Title.  '  Tariffa  De  Pefi  e  mefurc  cor-//refpondenti  dal  Lo 
uante  al  Ponete  :  da  // vna  terra  a  laltra  :  e  a  tutte  le  parte  del  // 
mondo  :  con  la  noticia  delle  robe  //  che  fe  trazeno  da  vno  Paefe 
//per  laltro.  Nouamente //co  diligentia  Ri-//ftampata  .•.►!*' 
(F.  i,r.) 

Colophon.  'CFiniffe  il  prohemio  de  //  Miffer  Bartholomio  di  // 
Pafi  da  Venefia.//  Finis.'     (F.  2 1 8,  v.) 

Descriptio7i.  8°,  10.2  X  15.1  cm.,  printed  in  double  columns, 
eachbeing4. 1  X  12.5  cm.  2  18  ff.  numb.,  30  11.  S.  1.  a.  (Venice  .?, 
c.  1510.) 

See  p.  77.  This  is  the  second  edition  of  this  popular  '  Tariffa.'  Since 
it  was  one  of  the  first  books  of  its  kind  to  appear  in  Venice,  its  five 
editions  are  easily  explained. 

ANTON    BARTHOLOMEO   DI    PAXI. 

Ed.  pr.  1503.  Venice,  1540. 

See  p.  77. 

Title.  '  Tariffa  //  de  i  pesi,  e  misvre  //  corrifpondenti  dal  Le- 
uante  al  Ponente  :  //  e  da  una  terra,  e  luogo  allaltro,  qua  fi  p  // 
tutte  le  parti  dil  Mondo  :  con  la  dichia-//ratione,  e  notificatione 
di  tutte  le  robbe  :  //  che  ft  tragono  di  uno  paefe  per  laltro.// 
Composta  per  M.  Bartholomeo  di  //  Paft  da  Vinetia.  Con  la  fua 
//Tauola  copiofifftma,  e  faci-//liffima  a  trouare  ogni  cofa//per 
ordine,  nuouamete // fatta  :  e  con  fomma // diligeza  reuista,//e 
stapata.// In  Vinetia.     M.D.  XL.'    (F.  i,  r.) 

Colophon.  '  In  Vinegia.  Nelli  cafe  di  Pietro  di  Nicolini  da 
Sabbio.//Ne  glianni  dilla  falutifera  Circoncifione  dil  no-//ftro 
Signore.     M.  D.  XL.// Dil  mefe  di  Genaio."    (F.  212,  r.) 

Description.  8°,  10. 3  X  14.7  cm.,  printed  in  double  columns, 
each  being  3.5  x  12.3  cm.  11  ff.  unnumb.  +  i  blank  +  200 
numb.  =  212  ff.,  30  11.    Venice,  1540. 

See  p.  77. 


8o  RARA   ARITHMETICA 

ANTON    BARTHOLOMEO   DI   PAXI. 

Ed.  pr.  1503.  Venice,  1557. 

See  p.  77. 

Title.  The  title  page  is  practically  identical  with  that  of  the 
1 540  edition,  except  for  the  date :  '  In  Vinegia  per  Paolo Gherardo. 
//M.  D.  LVII.'     (F.  I,  r.) 

Colophon.  '  In  Vinegia  per  //  Comin  da  Trino.//  M.  D.  LVII.' 
(F.  numbered  200,  r.) 

Description.  8°,  10.6  X  15  cm.,  printed  in  double  columns, 
each  being  3.7  X  12.3  cm.  1 1  ff.  unnumb.  +  200numb.  =21 1  ff., 
30  11.     Venice,  1557 

See  p.  77. 

BOETHIUS,  JODOCUS   CLICHTOVEUS,  and  FABER 
STAPULENSIS.     Ed.  pr.  1503.  Paris,  1503. 

For  the  biographies  of  Boethius  and  Faber  Stapulensis  see  pp.  25,  62. 
JoDOCUS  CLICHTOVEUS  was  bom  at  Nieuport,  Flanders;  died  at  Chartres, 
September  22,  1543.  He  was  educated  at  the  Sorbonne,  and  was  canon 
of  Saint-Jean,  at  Chartres.  Like  Faber  Stapulensis,  he  was  known  chiefly 
as  a  commentator. 

Title.  '  In  hoc  libro  contenta  //  Epitome/  copendiofaq5  // 
introductio  in  libros  //  Arithmeticos  diui  Seuerini  Boetij  :  adie// 
cto  familiari  cometario  dilucidata.//  Praxis  numerandi  certis 
quibufdam  re-//gulis  coftricta.//  Introductio  Tgeometria :  ex 
libris  diftlcta  //  Prim9  de  magnitudinib9  &  earu  //  circuftantiis.// 
Secudus  de  cofequentibus/  conti-//guis/  &  cotinuis.//  (Sur- 
rounded by  an  elaborate  woodcut.)  CTertius  de  puctis.  CQuar- 
tus  de  lineis.  CQuItus  de  fuperficieb9.//  cSextus  de  corporibus. 
CLiber  de  quadratura  circuli.  CLiber  de  cubica//tione  fphere. 
CPerfpectiua  introductio.     Clnfuper  aftronomicon.'     (F.  i,  r.) 

Colophon.  '  Id  opus  imprefferut  Volphgangus  //  hopilius  et 
Henricus  ftephanus  //  ea  in  arte  focii  in  Almo  pari-Z/fiorum 
ftudio  Anno  Chri//fti  Celorum  totiufqs //nature  cdditoris.// 
1503.    Die  vice//rimafepti-//ma  luV/nij.'     (F.  cxi,  v.) 

Description.  FoL,  19x26.6  cm.,  the  text  being  16x22.6 
cm.     112  ff.  numb.,  47-54  H-     Paris,  1503. 


PRINTIU)    IUX)KS  8i 

Editions.  This  is  the  first  edition  of  tliis  combination  of 
works,  the  second  (somewhat  changed)  ai)pearing  at  Paris  c. 
1 507  (see  below),  and  the  third  ib.,  i  5  10.  The  epitome  appeared 
as  a  'Compendium  arithmetices  Boethii,'  s.  1.,  in  1480,  and, 
with  the  arithmetic  of  Jordanus,  at  Paris,  in  1496.  There  was 
an  edition  of  Faber  Stapulensis,  Clichtoveus,  and  others,  at 
Cologne,  c.  15 15,  4°.  Scheubel  published  an  edition  at  Basel 
in  1553,  and  a  work  entitled  '  Arithmetica  Boethi  epitome  acced. 
Christian!  Morisani  Arithmetica'  also  appeared  at  Basel  in  1553 
(pp.  27,  182,  260).  The  'Praxis  numerandi '  of  Clichtoveus, 
of  which  this  is  the  first  edition,  appeared  separately  at  Paris  in 
15  10,  fol. 

The  copy  here  described  is  bound  with  the  1509-10  edition  of 
Bovillus  (see  p.  89).  It  consists  of  a  brief  introduction  by  Faber 
Stapulensis,  and  the  arithmetic  of  Boethius  with  the  commentary  of 
Clichtoveus  on  Faber's  epitome.  This  is  followed  by  the  geometry  and 
perspective  of  Faber  Stapulensis. 

BOETHIUS,    JODOCUS    CLICHTOVEUS,    and    FABER 
STAPULENSIS.     Ed.  pr.  1503.  Paris,  c.  1507. 

See  p.  80. 

Title.  'Introductio/Zjacobi  fabri  Stapulefis  in  Arithme//cam 
Diui  Seuerini  Boetij  pariter  j  Jordani//Ars  fupputadi  tarn  per 
calcu-//los  q3  notas  arithmeticas  fuis  quidem  regulis  elegater 
expreffa // Judoci  Clichtouei  Neoportuenlis.// Queftio  baud 
indigna  de  numeroru//et  p  digitos  :  particulos  finita  pgref'fione 
ex  Aurelio  Auguftino  //  CEpitome  rerum  geometricaru5  ex 
Geometrico  introductorio // Caroli  Bouilli.// CDe  quadratura 
Circuli  Demonftratio  ex  Campano.'     (F.  i,  r.) 

Description.  4°,  12.6  X  17.4  cm.,  the  text  being  10. 6  X  14.5 
cm.  32  ff.  unnumb.,  45-46  11.  Paris,  c.  1507.  The  dedicatory 
epistle  is  dated  '  Data  ano  falutis  //noftre  Millefimo  qngetefmio 
feptimo  tercio  caledas  lunij,'  that  is,  1507.  It  was  evidently 
printed  at  Paris,  but  it  is  without  date  or  place. 

Editions.    See  above. 


82  RARA   ARITHMETICA 

The  introduction  by  Faber  Stapulensis  to  the  arithmetic  of  Boethius 
and  Jordanus  was  very  popular  in  the  university  of  Paris  at  the  opening 
of  the  sixteenth  century.  It  is,  like  Boethius,  purely  theoretical.  The 
author  begins  with  a  dissertation  'de  vtilitate  arithmetice  difcipline,' 
and  then  gives  an  epitome  of  the  two  works.  This  is  followed  by  the 
'  compendium '  of  Clichtoveus,  merely  a  set  of  rules  for  the  operations. 
Books  of  this  character,  evidently  intended  as  the  bases  of  lectures  to 
university  students,  show  in  what  a  hopeless  state  the  Boethian  arith- 
metic found  itself  at  the  end  of  the  Middle  Ages. 

GREGORIUS   REISCH.     Ed.  pr.  1503.         Strasburg,  1 504. 

Bom  at  Balingen,  Wiirttemberg ;  died  at  Freiburg,  1523.  He  was  a  stu- 
dent at  Freiburg  in  1487,  and  took  his  bachelor's  and  master's  degrees  there. 
He  then  entered  the  Carthusian  order  and  became  prior  of  the  cloister  at 
Freiburg,  and  confessor  of  Maximilian  I. 

Title.  '  Aepitoma  omnis  phylosophiae.  ali-//as  Margarita  phy- 
losophica  tractans  //  de  omni  genere  fcibili :  Cum  additionibus  : 
Que  in  alijs  non  habentur.'  (Large  woodcut  representing  the 
liberal  arts.     F.  2,  r.    Plate  II.) 

Colophon.  '  Explicit  phylofophica  Margarita.  Caftigatione  acri 
//  In  nobili  Heluecioru  ciuitate  Argentina  Chalchogra-//phatu : 
Per  loanne  Gruninger  Ciue  Argetinii  :  I  vigilia  //  Mathie 
Anno  incarnationis  Saluatoris  M.ccccc.iiij.// Valete  &  Plaudite.' 
(F.  289,  V.) 

Description.  4°,  15.1  X  20  cm.,  the  text  being  1 1.5  X  15.6 
cm.  2  ff.  blank  +  289  unnumb.  =  291  ff.,  45  11.  The  illustra- 
tions are  hand-colored.     Strasburg,  1 504. 

Editiojts.  Freiburg,  1503,  4°;  Strasburg,  1504,  4°  (here 
described)  ;  Freiburg,  1504,  4°;  another  edition,  s.  1.,  by  Schott 
(Freiburg),  1504;  Strasburg,  1508;  Basel,  1508,  4°  (p.  83); 
ib.,  1512,  4°;  Strasburg,  1512,4°;  ib.,  1515,4°;  Basel,  1517, 
4°;  Paris,  1523  (first  Finaeus  edition);  Basel,  1535,  4°  (p.  84); 
ib.,  1583,4°;  Venice,  1594;  ib.,  1599;  ib.,  1600.  The  three 
Venetian  editions  (1594,  1599,  1600)  are  Italian  translations  by 
Giovanni  Paolo  Gullucci,  and  contain  the  additions  by  Orontius 
Finaeus,  and  also  the  introduction  by  Faber  Stapulensis  to  the 
arithmetics  of  Boethius  and  Jordanus  (see  p.  62),  Clichtoveus 


A.  Gf.()Mp:try 


B.    AkI  TUMI  lie 

Pi.ATK   II.    From    iiik  .m.\ki;\kii\   i'iiii.usoi'iiica 


PRINTED    BOOKS  83 

on  arithmetic  (sec  p.  80),  and  '  Questione  di  S.  A<;nstina  dclla 
progressione  dei  numeri  per  li  digiti,  et  per  li  arlicoli.'  I  lart- 
felder  {Zeitscli.  f.  Gcsch.  des  Obcrr/icitis,  II,  170)  has  sliown  that 
the  assertion  of  Hain,  Poggendorff,  and  others,  that  it  appeared 
in  1496,  is  incorrect. 

This  was  the  first  modern  encyclopedia  to  appear  in  i)rint.  It  contains 
a  compendium  of  the  trivium,  the  quadrivium,  and  the  natural  and  moral 
sciences.  It  is  made  up  of  twelve  books,  of  which  the  fourth,  consisting 
of  fifteen  folios  in  the  present  edition,  is  on  arithmetic.  The  author 
first  considers  the  definition  of  arithmetic,  and  then  gives  the  medi?eval 
classification  of  number,  including  the  system  of  ratios  as  set  forth  by 
Boethius  and  his  followers.  The  second  part  of  the  work  contains  a 
short  treatment  of  algorism,  including  the  fundamental  operations  and 
roots.  The  third  tractartus  relates  to  common  fractions  and  the  fourth 
to  physical  or  sexagesimal  fractions.  The  arithmetic  closes  with  a 
treatment  of  line  reckoning,  giving  the  four  fundamental  operations 
and  the  rule  of  three.  The  illustrations  are  particularly  interesting. 
(See  Plate  II.) 

Other  works  of  I^OJ.  Boethius,  p.  27,  1488;  Bradwardin,  p.  61, 
1495;  Faber  Stapulensis,  p.  62,  1496;  Huswirt,  p.  74,  1501  ;  Jor- 
danus,  p.  62,  1496;  Orbellis,  p.  23,  1485;  Peurbach,  p.  53,  1492; 
Sacrobosco,  p.  32,  1488;  Torrentinl,  p.  76,  1501  ;  Anonymous, 
'Textus  arithmetic3e  comunis,  cum  Conradi  Norici  commentatione,' 
Leipzig,  fol. 

Works  0/1504.  Anianus,  p.  35,  1488;  Bradwardin,  p.  61,  1495  ; 
Johannes  Carolus  (see  Landshut,  below)  ;  Huswirt,  p.  74,  1501  ;  Reisch, 
p.  82,  1503;  Sacrobosco,  p.  32,  T488  ;  Johann  Karl  von  Landshut 
(Lanzut),  '  Algorithmus  integrorum,'  Leipzig  (see  also  p.  97,  1513- 
1515)  ;  Henricus  Stromer,  '  Algorithmus  linealis  cum  Regula  de  Tri,' 
Leipzig,  4°,  with  other  editions  in  15 10;  15 12,  4°;  15 14;  I-^eipzig, 
1516,  4°;   ib.,  1517,  4°;    1520. 

GREGORIUS   REISCH.     Ed.  pr.  1503.  Basel,  1508. 

See  p.  82. 

Title.  '  Margarita  philofophica  //  cu  additionibus  nouis  :  ab 
auctore  fuo // ftudiofiffima  reuifioe  tertio  fup  additis.// Jo. 
Schottus  Argen.  lectori.  S.//  Hanc  emo/  non  preffam  mendaci 
ftigmate/ Lector  ://Pluribus  aft  auctam  pcrlcge  :  doctus  eris.// 
Bafilce.  1508  '     (P\  i,  r.) 


84  RARA   ARITHMETICA 

Colophon.  '  CTertio  induftria  complicu  Micha//elis  Furterij/ 
et  Joanis  ScotiZ/ftudiofiiffime  preffa.  Ba-/Ailee.i4.Kar.Mar// 
tias.    Anno  Chrifti.// 1508.'     (F.  308,  r.) 

Description.  4°,  15.2  X  21.9  cm.,  the  text  being  12.7  x  16.7 
cm.  309  ff.  unnumb.,  42  11.  In  this  edition  the  leading  initials 
are  inserted  by  hand  and  the  illustrations  are  colored.  Basel, 
1508. 

See  p.  83. 

GREGORIUS   REISCH.     Ed.  pr.  1503.  Basel,  1535. 

See  p.  82. 

Title.  '  Marga-//rita  philosophica,  rati-//onalis,  Moralis  phi- 
lofophiae  princi-//pia,  doudecim  libris  dialogice  c6ple-//ctens, 
olim  ab  ipfo  autore  recognita  :  //  nuper  aut  ab  Orontio  Fineo 
Delphi//nate  caftigata  &  aucta,  una  cum  ap-//pendicibus  itidem 
emedatis,  &  qua // plurimis  additionibus  &  figuris,  ab//eodem 
infignitis.  Quoru  omni-//um  copiofus  index,  uerfa //continetur 
pagella.//Virefcit  uulnere  uirtus.//Basileae  1535.'  (Surrounded 
by  an  elaborate  woodcut.)     (P.  i.) 

Colophon.  '  Basileae  excvdebat  Henricvs  //  Petrus,  ac  Con- 
radi  Refchij  impenfis.     An«//M.  D.  XXXV.'     (P.  1577.) 

Description.  4°,  15  X  20.8  cm.,  the  text  being  9.6  X  13.8  cm. 
78  pp.  unnumb.  +  1498  numb.  +  I  blank  =  1577  pp.,  26-30  11. 
Basel,  1535. 

Editions.  See  p.  82.  Finaeus  dates  the  dedicatory  epistle 
'  Parifijs  ex  regali  collegio  Nauarrae.  1523,'  and  his  first  edition 
appeared  in  that  year.  This  edition  gives  only  part  of  the  elabo- 
rate engravings  found  in  the  earlier  ones.  It  is,  however,  much 
better  printed,  being  set  in  clear  Roman  type  and  having  a  more 
open  page. 

See  p.  83. 

THEODOR   TZWIVEL.     Ed.  pr.  1505.  Cologne,  1507. 

A  German  arithmetician  of  c.  1500,  from  Monte  Gaudio  (Mongavensis), 
Westphalia. 

Title.    See  Fig.  46. 


PRINTIH)    lUJOKS  S5 

Colophon.  'CAli^orilhnii.  qui  ars  dicilur  mimerancli.  dc  intct^iis 
//  per  figurarum  (more  Alemanorum)  deletionej.  Nec//no  dc 
pportionib?  ingcniofi  Pythagorifte  Theodo//rici  Tzwyucl.  pofl 
plurima  praxin  iam  tande  in  hoc // opendiu5  reducti  finis  adcl'l 
quod  et  puplica  ob  vti//litatem  in  magiftrali  artis  imprcffori  e 
taberna  inge-//nuorum  liberorum  Quentell  iterato  diffeminari  pro 


WSmmt 

opufmlatiuo  ftieotiortri  timi 

ud  benumcroppjaiciCqiK  algo:itl?mi  fticutif )vnu  a  mtegns  per 
figuraru(mo:e  aUindnoy)&detion&BlCfl-uOe  p:opomonibii8  01 
tu8  vfas  frcQUcns  m  mndcl  (^armontca  Soicrini  :6oatj 

« 

De  vrilitatc  hums  libdli  Tetraftichort 
ioannisMurmcIIi]  RuremundenCs 

Si  quisarithmcdc^optatcognolcerc  praxin 
Pythagorenumejfosdifcerefi  quis  amat 

Sdrc  mathcriiaacch  fi  vult^fi  denic^  quic^ 
Oefophia*huncinodfcunicomparetere  libru 

FlC  46.     'lYrLK    PAGF.    OK     TZWIN  Kl, 

//curauit.  Anno  a  natali  dominico  Millefimo  quin//genctefuno- 
feptimo.'    (F.  9,  v.     See  Fig.  47.) 

Description.  4°,  13.9  X  20  cm.,  the  text  being  8.8  X  14.5  cm. 
10  ff.  (i  blank),  46  11.     Cologne,  1507. 

Editions.  Gunther  mentions  a  Miinster  edition  of  1505,  but 
I  have  not  seen  it ;  there  was  a  Cologne  edition  in  the  same 
year;  Cologne,  1507,  4°  (here  described). 

The  work  is  divided  into  two  parts,  the  first  beginning  as  follows  : 
'  <LAlgorithmiis  de   integris  p  figuranl  (more  alemano^)  cIeleti-//one 


86  KARA   ARITHMETICA 

arte  numeradi  enucleatim  ^pendiofeqj  edocens.'  (F.  2,  v.)  This  part 
contains  a  brief  explanation  of  the  writing  of  numbers  and  the  funda- 
mental operations.  The  second  part  begins  as  follows  :  '  <I.Algorithm9 
de  pportioib?  cuius  vfus  freques  in  muficam  har//monicam  Seuerini 
Boetij  '  (f.  8,  v.),  and  two  pages  treat  of  the  operations  with  the 
mediaeval  '  proportiones '  or  ratios,  A  comparison  of  the  title  pages 
represented  on  pp.  45,  75,  85,  and  of  the  works  to  which  they  belong,  at 
least  two  of  which  were  printed  in  Cologne,  leads  to  the  belief  that  the 
expression  '  per  figuraru  (more  alemanor^)  deletione '  (by  the  deletion 
of  figures  in  the  German  way)  refers  to  a  contemporary  North  German 
custom  of  not  actually  canceling  the  figures  in  the  galley  division,  as 
the  Italians  did. 

Other  works  0/1^05.  Anonymous  (see  Licht),  p.  70,  1500  ;  Borghi, 
p.  16,1484;  Bradwardin,  p.  61,  1495  ;  Ciruelo,  p.  60, 1495  ;  Licht,  p.  70, 

1500;  Suiseth,p.  10,  c.  1480;  Tor- 
faiao:itl?mrquiarebictturntimerandi.KmwgTi8  rentini,  p.  76,  Moi  :  Anonymous, 
p:rfiaurarum(tno2caicm3nozum)a:ictioiicj.  flee  ^     '  ^  .  ^ 

n5«ppo:tionibDingcniofi-pytI?3go2i(]£Z:i?codo     *  Tractatus     perutihs    in    anthme- 
nciiiwyucl.podplurimapuiciiiumtatidcitiboc      .■        cnprnlativa  '    Paris-     Hprman 
ppEnd.uutducnfiniafldcft.Quodctpuplicaobvti     ^^^^  specuiativa,    raris,    Merman- 
liHwminm3fii(lraliarn8itnp;elTo:ictalxriiaingc/      nus  Buschius,    'Enchiridion  novus 
curau.t.  aftnoanataUDominiwZTPaUfimoqmii     Algorismi,  Cologne,  s.  a.  (i  504),  4 
fioitcrimofcpumoi  (see  also  p.  106,  15 14);   Nicolaus 

Fig.  47.    Colophon  of  tzwivel      Horem,    'Tractatus    proportionu,' 

Venice,  in  a  volume  with  Bradwar- 
din and  Suiseth  (see  also  p.  1 16,  15 15)  ;  '  Algorithmus  linealis  Baccalariii 
Wolfgangum  Monacenseni,'  Leipzig,  4°  ;  Georg  Leunbach,  an  arithmetic. 

Works  of  1506.  Albert  of  Saxony,  p.  9,  c.  1478  ;  Anonymous,  p. 
10,  1480;  Vittori,  p.  9,  Albert  of  Saxony,  c.  1478,  and  p.  61,  Brad- 
wardin, 1495;  Pietro  Borriglione,  '  Arismetices  praxis,'  Turin,  22  fif., 
with  a  second  edition,  ib.,  1523  ;  Raphael  Maffei, '  Commentarii  Urbani,' 
Rome,  fob,  an  encyclopedia  containing  a  book  (no.  35)  '  De  scientiis 
mathematicis,'  which  includes  a  little  arithmetic  ;  Maffei's  work  was 
also  printed  in  1527,  and  at  Paris  in  1511,  1515,  1526,  and  1530,  and 
at  Basel  in  1559. 

ANONYMOUS.     Ed.  pr.  1507.  Leipzig,  1507. 

Title.    See  Fig.  48. 

Colophon.  '  Cimpreffum  Liptzck  per  Baccalariu  Vuolfgangu 
// Monacenfem  Anno  noftre  redemptionis  .  1507.'     (F.  27,  r.) 

Description.  4°,  14.5  X  19.5  cm.,  the  text  being  9.5  x  15.3 
cm.    28  ff.  unnumb.  +  i  blank  =  29  ff.,  28-34  11.     Leipzig,  i  507. 


PRINTED    BOOKS  87 

Editions.  Leipzig,  1507  (here  described) ;  ib.,  1509,4°;  s.  1.  a. 
(Nurnberg  ?,  c.  15  10). 

This  resembles  several  of  the  works  on  algorism  appearing  about  this 
time,  such  as  Widman's  (?),  Licht's,  and  Huswirt's.  It  contains  a  very 
brief  treatment  of  the  fundamental  operations,  including  duplation  and 
mediation.  In  division,  only  a  single  example  is  given,  that  of  1456  -=- 
12.    After  a  similarly  brief  treatment  of  fractions,  the  Kuethian  jjropor- 

Fig.  48.    Title  of  the   1507  Ah^orithmus 

tions  (ratios)  are  taken  up  :  '  Sequitur  Algorithmus  proportionii.'  Then 
follow  the  Rule  of  Three  and  several  other  rules  now  entirely  forgotten, 
such  as  '  Regula  legis,'  '  Regula  augmenti,'  '  Regula  plurima,'  '  Regula 
pulchra,'  and  '  Regula  falfi.'  Such  '  regulae  '  were  not  stated  like  modern 
rules  but  consisted  of  groups  of  similar  problems. 

Other  xvorks  of  1^07.  Boethius,  p.  27,  1488  ;  Faber  Stapulensis,  c. 
1507,  p.  81,  1503  ;  Huswirt,  p.  74,  1501  ;  Peurbach,  p.  53,  1492  ;  Tor- 
rentini,  p.  76,  1501. 

Works  of  1^08.  Anianus,  p.  32,  1488;  Reisch,  p.  82,  1503  ;  Widman, 
p.  39,  1489  ;  Hieronimus  de  Hangest,  '  Liber  proportionum,'  Paris,  4°. 

LUCA   PACIUOLO.     Ed.  pr.  1509.  Venice,  1509. 

See  p.  54. 

Title.  After  a  vocabulary  and  index  the  work  begins  on  f. 
I,  r.,  as  follows  :  '  Excellentiffimo  principi  Ludouico  mariae  Sfor. 
Anglo  Mediolanen/Zfium  duci:  pacis  et  belli  ornamento  fratris 
Lticae  pacioli  ex  Burgo  fancti  //  Sepulchri  ordinis  Minorum:  Sa- 
cras  theologiae  pfefforis.  Ue  diuina  pro//portione  epiftola.' 


RARA   ARITHMETICA 


Fig.  49.    From  the   1507  A/go rifJi nuts 


PRINTED    BOOKS  89 

Colophon.  'CVcnetiis  Impreffum  per  probvini  viruni  Paganinum 
de  paganinis  de  //  Brixia.  Decrcto  lamcii  i)uhlic()  vt  nullus  ibidem 
totiqj  dominio  annorum  //  XV.  curicult)  imprimat  vel  Tprimcre 
facial.  Et  alibi  impreffum  fub  quouis // colore  in  publicum  ducat 
fub  penis  in  dicto  priuilegio  contentis.  Anno  Re//demptionis 
noftre.M.D. Villi.  Klen.Iunii.  Leonardo  LauretanoVe.//Rem.?u. 
Gubernante.  Pontificatus  lulii.II.  Anno.VI.'  (F.  27,  r.,  of  Part  3. 
A  similar  colophon  appears  at  the  foot  of  f.  35,  v.,  of  Part  i.) 

Description.  8°,  20.5  X  28.3  cm.,  the  text  being  9.1  X  21.5 
(with  marginal  drawings),  i  f.  blank  +  2  unnumb.  +  90  numb. 
93  ff.,  52-57  11.     Venice,  1509. 

Editions.    There  was  no  other  edition. 

With  some  hesitancy  this  book  has  been  included,  the  only  justifica- 
tion being  the  fact  that  there  are  several  pages  devoted  to  the  discussion 
of  proportion  in  general,  including  the  arithmetical,  geometric,  and 
astronomic.  Paciuolo  excludes  the  other  forms  of  ancient  proportion 
given  by  '  Platone  e  Arifto.  e  yfidoro  i  le  fue  ethimologie.  El  feuerin 
Boetio  in  fua  arithmetica.'  (F.  5,  r.)  Most  of  the  treatise  is,  however, 
devoted  to  geometry. 

CAROLUS   BOVILLUS.    Ed.  pr.  1509-10.    Paris,  1509-10. 

BouvELLES,  BoiJELLES,  BouiLLES,  BouvF.i,.    Born  at  Saucourt,  Picardy, 
c.  1470 ;  died  at  Noyon,  c.  1 553.    Canon  and  professor  of  theology  at  Noyon. 

Title.    See  Fig.  50. 

ColopJion.  '  Libelli  De  Mathematicis  Svpple-//mentis  Finis 
Anno  Salutis  Humane// 1509  lanuarij  Die  Deci//maoctaua // 
CEditum  est  vniversvm  hoc  volvmen  Ambianis  in  edibvs  Re// 
uerendi  in  Christo  Patris  Francifci  De  Hallevvin  Eiufdem  Loci 
Pontificis  .-.  //  Et  emiffum  ex  officina  Henrici  ftephani.  Impenfis 
eiufdem  et  loannis  parui  in  chalcotypa //arte  fociorum  Anno 
Christi  Saluatoris  omnium  i  5  10.  Primo  Cal.  Februarij.//  Parisiis.' 
(F.  198,  V.) 

Description.  FoL,  19.7  X  27.4  cm.,  the  text  being  15.1  X  25.8 
cm.     198  ff.  numb.,  53-54  11-     Paris,  1509-10. 

Editions.  Fontes  has  described  (Toulouse  ]\Iem.  (9)  VI,  155- 
167,  for   1894)  a  quarto  of    15 10,  published   by   Stephanus  of 


([InrupmatliematfcuopiisquadripartitudCDcNumensPerfeaisll^Dc 
Mathcmaticis  Rofis  ^Ot  Geometricis  Gorporibus 
(XDc  Geometrids  Supplemcntis 

Fig.  50.   Title  page  of  bovillus 


PRINTED    BOOKS  91 

Paris,  enlilled,  *  Caroli  Bovilli  lihcr  do  numeris  pcrfcclis.'  This 
book  is,  however,  without  title  pa^^e,  and  is  of  course  sinij)!)-  ff. 
172-180  of  this  work.     I  know  of  no  other  editions. 

The  first  part  on  numbers,  the  '  Lil)cr  de  duodecim  numeris,'  begins 
on  f.  148,  v.,  and  ends  on  f.  171,  r.,  with  the  colophon  :  '  I.ibri  dvodecim 
nvmerorvm  finis  :  editi  //  in  dome-  R.  P.  Francifci  de  Hallevvin/  Pon- 
tificis  Ambi//anenfis.  Anno  ab  autore  numerorum  Incarnate.// 15 10  : 
Mai]  decimafexta.'  It  relates  solely  to  the  mystery  of  numbers  and  to 
the  Greek  theory.  The  part  on  perfect  numbers,  numbers  which  equal 
one  plus  the  sum  of  their  factors,  begins  on  f.  172,  r.,  and  ends  on  f. 
180,  r.,  with  the  following  colophon  :  '  CLiber  perfectorvm  nvmerorvm 
finis.//  Perfecto/  trinoqj  deo  laudes  in  gentes.  Anno  domi-//ni/  1509  : 
lanuarij  4.'  The  dates  of  the  several  parts  vary  from  October  25,  1509, 
to  January  18,  1509  (1510  N.  S.). 

Other  works  0/  ijog.  Anianus,  p.  32,  1488;  Anonymous,  p.  87, 
1507  ;  Borghi,  p.  16,  1484  ;  Ciruelo,  p.  60,  1495  ;  Licht,  p.  70,  1500; 
Sacrobosco,  p.  32,  1488. 

Other  works  of  ijio.  Anonymous,  \).  87,  1507  ;  Anonymous,  c. 
15  10,  p.  46,  1491  ;  Boethius,  p.  30,  1488  ;  Bovillus,  p.  89,  1509  ;  Brad- 
wardin,  c.  15 10,  p.  61,  1495  ;  Clichtoveus,  p.  81,  1503  ;  Faber  Stapu- 
lensis,  p.  62,  1496  ;  Jordanus,  p.  62,  1496;  Peurbach,  p.  53,  1492  ; 
Sacrobosco,  p.  32,  1488  ;  Stromer,  p.  83,  1504  ;  Paxi,  c.  15  10,  p.  79, 
1503  ;  Torrentini,  p.  76,  1501. 

Works  of  i^Ji.  Anianus,  p.  32,  1488;  Boethius,  p.  27,  1488; 
Maffei  (Maphjeus),  p.  86,  1506  ;  Peurbach,  p.  53,  1492  ;  Simon  Eisen- 
mann,  'Enchiridion  arithmetica,'  J>eipzig,  fol. 

JUAN   DE  ORTEGA.     Ed.  pr.  15 12.  Rome,  1515. 

John  de  Lortze.    A    Spanish    priest    of   the    Dominican    order,    from 
Aragon.    He  was  still  living  in   1567. 

Title.  '  Svma  //  de  Arithmetica  :  Geometria  //  Pratica  vtiliffi- 
ma:  ()rdina//ta  per  Johane  de  Or//tega  Spagnolo//Palentino.// 
Cum  Priuilegio.'    (Surrounded  by  an  elaborate  v^oodcut.   F.  i,  r.) 

Colophon.  '  Impreffo  in  Roma  per  Maftro  Stephano  Guillen 
de  Lorena//anno  del  noftro  Signore  15  15  adi  10  de  Nouebre 
Regnante  Leo//ne  Papadecimo  in  fuo  Annotertio.'    (F.  1 16,  r.) 

Description.  Fol.,  20.6  X  30.2  cm.,  the  text  being  13.1  X  22.7 
cm.  2  ff.  unnumb. +  114  numb.  =  116  ff.,  32-38  11.  Rome, 
1515. 


RARA  ARITHMETICA 


From  the  15 15  ortega,  introduction  to  barter 


i 


PRINTED    liOOKS  93 

Editions.  Barcelona,  15 12;  Lyons,  1512,  4";  ih.,  1515,4°; 
Rome,  15 15,  fol.  (here  described);  Messina,  1522;  1534; 
Seville,  1536;  ib.,  1537;  Paris,  1 540  (?)  ;  Seville,  1542,  4° 
(see  below);  ib.,  1552  (p.  94);  s.  1.  (?),  1552;  Granada,  1563, 
4°.  The  Lyons  edition  of  15 12  was  the  first  book  on  commer- 
cial arithmetic  printed  in  France.  It  differs  somewhat  from 
the  Rome  edition  of  15  15,  but  the  latter  differs  only  a  lillle 
from  the  first  (Barcelona)  edition.  Mr.  Plimpton's  copy  of  i  5  i  5 
belonged  to  Prince  Boncompagni  and  has  his  collation  on  the 
cover.  It  is  beautifully  printed  and  is  one  of  the  best  examples 
of  the  early  Italian  mathematical  typography. 

This  is  one  of  the  most  celebrated  arithmetics  written  in  Spain  in 
the  sixteenth  century.  It  is  a  purely  commercial  book,  beginning  with 
notation,  taking  up  the  four  processes  with  integers,  the  progressions, 
the  roots,  and  the  checks  on  operations,  and  the  same  operations  in 
the  same  order  with  fractions,  and  then  discussing  the  business  rules. 
These  last  include  exchange,  rule  of  three,  profit  and  loss,  partnership, 
testament  problems,  barter,  alloys,  false  position,  and  a  little  mensuration. 

Other  works  0/1512.  Bradwardin,  p.  61,  1495  ;  Peurbach,  p.  53, 
1492  ;  Reisch,  p.  82,  1503  ;  Stromer,  p.  83,  1504  ;  Torrentini,  p.  76, 
150 1  ;   I.  Furst,  '  Novus  .  .  .  algorithmus.' 

JUAN    DE   ORTEGA.     Ed.  pr.  15 12.  Seville,  1542. 

See  p.  91. 

Title.  '  Tratado  //  fubtiliffimo  de  Arifmeti-//ca  y  dc  Geome- 
tria  :  co-//puefto  y  ordenado  //  por  el  reuerendo  //  padre  fray  Jua 
//  de  Ortega  d'  //la  orden  d'  //  los  pre//dicadores.//  1542  // 
1234567890.'  (The  whole  is  surrounded  by  an  elaborate  wood- 
cut border.)     (F.  i,  r.) 

Colophon.  'Sue  impreffo  el  prefente  libro // re  Arifmetica  y 
Geometria  (agora  nueuamete  //  corregido  y  emendado)  en  cafa 
d'Jacom//cr6berger:  enla  muy  noble  y  muy  leal//ciudad  de 
Seuilla:a  cinco  dias//de  deziembre  de.  M.d.  y //quareta  y  dos 
ahoz.'     (F.  232,  V.) 

Description.  4°,  14.5  X  20.5  cm.,  the  text  being  11.3  X  17.2 
cm.     232  ff.  numb.,  34  11.     Seville,  1542. 

See  above. 


94  RARA   ARITHMETICA 

JUAN    DE   ORTEGA.     Ed.  pr.  1512.  Seville,  1552. 

See  p.  91. 

Title.  '  Tractado  //  Subtiliffimo  d  Arifmetica  y  de  Geome// 
tria.  Compuefto  por  el  reueredo  padre  //  irdij  Juan  de  Hortega, 
d'la  orden // de  los  predicadores.// Elhora  de  nueuo  emendado 
con  mucha // diligetia  por  Gon^alo  Bufto  d'muchos //errores 
que  aula  en  algunas  im//preffiones  paffadas.//  CVan  anadidas 
en  efta  impreffion  las  //  prueuas  defde  reduzir  hafta  partir  que- 
//brados.  Y  en  las  mas  de  las  figuras  de  //geometria  fus  prue- 
uas, con  ciert  os  aui-//fos  fubjetos  al  Algebra.  Y  al  fin  defte  tra 
//ctado  :  13.  exemplos  de  arte  mayor.// 1552.'  (Title  page  is 
printed  in  red,  and  is  surrounded  by  an  elaborate  woodcut  in 
black.)     {F.  I,  r.) 

ColopJion.  '  Hizo  fin  el  tractado  de  Arifmetica  Y// Geome- 
tria, que  compufo  y  ordeno  el  reuerendo  padre //fray  Juan  de 
Hortega,  de  la  orden  de  los  predica-//dores.  Fue  impreffo  ela 
muy  noble  :  muy  leal  //  ciudad  de  Seuilla  por  Jua  canalla,  enla  // 
collacion  de  fant  Jua.  Acabofe  a  diez//y  feys  dias  del  mes  de 
Abril  del  //  ano  de  nueftro  criador  y  rede//ptor  Jefu  Christo  de 
mill// 2  quinientos  2  cin-//quenta  y  dos//aiios.'    (F.  223,  r.) 

Description.  8°,  14.6  X  20  cm.,  the  text  being  1 1.9  x  17.4  cm. 
232  ff.  numb,  -f  7  unnumb.  =  239  ff.,  33  11.     Seville,  1552. 

See  p.  93.    The  'arte  mayor'  mentioned  in  the  title  is  algebra. 

JODOCUS   CLICHTOVEUS.     Ed.  pr.  1513.        Paris,  1513. 

See  p.  80. 

Title.    See  Fig.  52. 

Colophon.  '  CExpletum  eft  hoc  opufculum  &  ex  ofificina  emif- 
//fum/  in  alma  Parifiorum  academia  :  ano  domi-//ni  (qui  omnia 
numero  definiuit)  decimoter//tio  fupra  millefimu  &  quingente- 
fimu/ //decimafexta  die  Decembris.  Per // Henricu  ftephanu/ 
artis  excu//forie  libroru  fedulu  &  indu//ftriu  opifice/  e  regione 
//fchole  Decretoru //habitan-//tem.'     (F.  43,  v.) 

Description.  4°,  14  X  19.5  cm.,  the  text  being  12  x  15.7  cm. 
41  ff.  numb.  +  3  unnumb.  -f-  2  blank  =  46  ff.,  42  11.     Paris,  15  13. 


PRINTED    HOOKS  95 

Editio)is.    There  was  no  other  edition. 

This  is,  I  believe,  the  first  separate  treatise  on  the  mystery  of  numbers 
to  appear  in  print.  Paciuolo  had  incUided  a  good  deal  of  such  material 
in  his  Siniima  of  1494,  and  about  a  century  later  Bungus  published  a 
monumental  treatise  upon  the  subject,  but  Clichtoveus  was  a  pioneer 
in  the  publication  of  a  separate  work.    The  result  of  his  labors  is  properly 

ltDc  myflica  numcrorumfignifi'cationc  opu^ 

fculumteorum  pr^fertim  qui  in  facris  lit* 

teris  vfitati  habentur/rpirituale  ipfoi» 

rum  defignationem  (uccin » 

{Xz  elucidanst 

%♦ 

Fig.  52.   TrrLE  page  of  the   15 13  clichtoveus 

included  in  a  list  of  arithmetics,  for,  while  there  is  nothing  of  compu- 
tation in  the  work,  it  is  not  unrelated  to  the  number  theories  of  the 
mediaeval  writers  and  even  of  the  Pythagoreans. 

Clichtoveus  discusses,  as  is  usual  among  such  writers,  the  religious 
significance  of  one  ('  Quid  vnitas/  numerorum  fons  et  origio  defignat. 
Cap.  I.')  and  the  numbers  of  the  first  decade.  He  also  mentions  several 
larger  numbers  which  were  supposed  to  have  some  scriptural  significance, 
not  forgetting,  of  course,  666,  '  the  number  of  the  beast.' 

There  is  also  in  this  work  a  chapter,  generally  unrecognized  by  writers 
on  the  history  of  the  subject,  on  finger-reckoning  :  '  Quomodo  antiqui: 
numeros  omnes  per  certas  digitorii  &  manuum  figurationes/  fignificare 
iunt  foliti.'     Cap.  XXVIII. 

JOANNES   MARTINUS   BLASIUS,  Villagarciensis. 

Ed.  pr.  15  13.  Paris,  15  13. 

A  Spanish  astrologer  and  arithmetician  of  c.  1500.  In  this  edition  the 
author's  name  appears  as  '  loannes  Martinus  Rlafius  diocefis  Pacefis,'  and 
in  the  1519  edition  as  'loannes  Martinus  Silecevs  (and  Sciliceus)  Diocesis 
Pacefis.' 

Title.    See  Fig.  53. 

Colophon.  'Explicit  liber  Arithme//tices  practice  magri  Joan- 
nis  Martini  Blafij  Vil-Z/lagarciefis  :  Parifijs  edit9  in  honeftilTima 
Belua-//coru  paleftra  :  impreffus  vero  a  calcographorum  ex-// 
pertiffimo  Thoma  Kees  :  Vvefalienfe  expenfis  pro//birrimorimi 


KARA   ARITHMETICA 


Fig.  53.   Title  page  of  the   15 13  blasius 


PRINTr:i)    IKJOKS  97 

vironim  :  Joannis  Parui  cl  Joannis // Lambert.  Anno  doiiiini. 
15  13.  in  vigilia  diui  Jo-//annis  baptifte.'     (F.  26,  r.) 

Description.  Fol,  19.5  X  28.  i  cm.,  printed  in  double  columns, 
each  being  6.5  X  21.3  cm.    26  ff.  unnumb.,  64-66  11.    Paris,  1513. 

Editions.  Paris,  15 13,  fol.  (here  described);  ib.,  1514,  large 
8°;   ib.,  15  19  (see  below);  ib.,  1526,  fol. 

Although  an  algorism,  the  work  is  mediaeval  in  character.  The  author 
first  discusses  the  fundamental  operations  with  integers,  including  series 
and  roots  as  was  the  custom,  but  not  considering  duplation  and  media- 
tion as  distinct  topics.  He  is  one  of  the  earliest  writers  to  adopt  the 
spelling  substractio,  for  subtraction,  a  custom  more  or  less  followed  by 
the  Dutch  and  English  arithmeticians  for  several  generations.  Blasius 
closes  his  '  primus  tractatus '  with  a  discussion  of  compound  numbers. 
'  Tractatus  secundus  '  considers  computations  with  counters,  or  '  nummi 
supputatorii.'  The  '  tertius  tractatus  '  is  devoted  to  common  fractions, 
'  fractiones  vulgares  ' ;  the  '  quartus  tractatus  '  to  sexagesimal  fractions, 
*  fractiones  ])hisicae ' ;  and  the  '  quintus  tractatus '  to  the  rule  of  three 
('  Prima  regula  1  fudametalis  qua  detri  dicunt.')  There  are  no  practical 
applications  of  any  value. 

This  first  edition  differs  greatly  from  the  third  (15 19)  described 
below.  It  includes  only  the  algorism,  while  the  latter  consists  of  two 
parts,  the  first  l)eing  on  Boethian  arithmetic,  and  the  second  being 
substantially  identical  with  the  15 13  edition. 

Other  7uorks  of  15 13.  Albert  of  Saxony,  p.  9,  c.  1478;  Ciruelo, 
p.  60,  1495  ;  (ryraldus,  p.  254,  1553  ;  Licht,  p.  70,  1500  ;  Peurbach,  p. 
53,  1492  ;  Johann  Karl  von  Landshut  (Lanzut),  'Algorithmus  linealis,' 
Cracow,  with  editions,  ib.,  15 15,  15  19,  4°  (see  p.  83,  1504). 

JOANNES   MARTINUS   BLASIUS,  Villagarciensis. 

Ed.  pr.  I  5  13.  Paris,  15  19. 

See  p.  95. 

Orontius  FiNARtrs,  editor.    See  p.  i6o. 

Title.  '  Arithmetica // loannis // Martini,  Scili-//cei,  in  the- 
oricen,  et  praxim  //  fciffa,  nuper  ab  Orontio  Fine,  Del//phinate, 
fumma  diligcntia  caftigata,  lon-//geq3  caftigatius  q  prius,  ipfo 
cura-//te  impreffa  :  omni  hominij  //  conditioni  perq  //  vtilis,// 
&  neceffaria.// Virefcit  vulnere  virtus.'  Surrounded  by  an  elab- 
orate woodcut,  with   the  following  on  four  sides  :   '  Emissa  ex 


98  RARA   ARITHMETICA 

officina  Ilenrici  Stephani,  e  regione  //scholae  Decretorvm  Com- 
morantis,// vbi  et  vaenalis  reperitvr.// Parisiis  anno  Christi. 
1519.')     (F.  I,  r.) 

Description.  Large  8°,  20.3  X  28.1  cm.,  the  text  being  13.3  x 
20  cm.     64  ff.  numb.,  52  11.     Paris,  15  19. 

Editions.    See  p.  97. 

The  first  half  of  this  rare  work,  not  found  in  the  15 13  edition,  is 
one  of  the  best  exponents  of  the  Boethian  arithmetic  of  the  time. 
Finaeus,  the  editor,  refers  to  the  author's  work  in  these  words :  '  Hanc 
loanes  Martinus,  Sciliceus,  Hifpanus,  vir  Mathematicarii  peritus,  noftra 
tepeftate  Parifijs  edidit.'  The  author  shows  a  good  knowledge  of  the 
ancient  writers,  mentioning  particularly  Pythagoras,  Nicomachus,  Euclid, 
Apuleius,  and  Boethius,  together  with  Jordanus,  Faber  Stapulensis,  and 
Clichtoveus.  The  distinctive  superiority  of  this  part  of  the  work  lies  in 
the  clearness  and  arrangement  of  the  illustrations  of  the  various  classes 
of  numbers  defined.  The  theory  of  numbers  ends  on  f.  24,  v.,  the  prac- 
tical part  beginning  on  f.  25,  v.  Among  the  most  noteworthy  features 
of  this  half  of  the  work  is  a  '  Tabvla  mvltiplicationis  et  divisionis '  with 
all  products  to  50  x  50. 

RAGGIUS  FLORENTINUS.    Ed.  pr.  15 14.    Florence,  1520. 

A  Florentine  mathematician  of  the  fifteenth  century. 

Title.     See  Fig. 


UJ  Inhocopufculoheccontinentur, 


54- 


CQaidGtproportib&quoteiusffedcs  Colophon.  'Cim- 

preffum    Floretiae 


Bernardum  Zuc- 
chetta  //  Anno. 
M.D.XX.      lanua- 

|i;Qucmatorniinoruct>roportiodlccndant  "J-  ^^•'  (F- 1 1 ,  v.) 

Description.  4°, 
13.4  x  20.1  cm., 
the  text  being  9.2 

CConfutationesaigumentorum  calculator^  x  1 5  cm.     1 1   ff . 

T.  rr.  unnumb.,  34-37 11. 

Fig.  54.    Title  pace  of  raggius  ' -^^   ■^' 

Florence,   1520. 

Editions.    Florence,   1514,4°;   ib.,   i  520,  4°  (here  described). 


^  Quo  intelle<fhi  compolitio  S:  diuilio  proport/onum 
acclpiatur  &  niale  opinantium  confutationes 


CTQuidptopinquftas  &•  rcmotio 


PRINTED    BOOKS  99 

This  work  consists  of  a  theoretical  treatment  of  proportion.  While 
partly  arithmetical,  this  treatment  relates  to  the  fundamental  theory, 
and  is  equally  applicable  to  geometry.  The  book  is  dedicated  to  the 
illustrious  Ciiovanni  Salviati,  uncle  of  Cosimo  I,  Cirand  Duke  of  Tuscany. 

GUILLIELMUS    BUDAEUS.   Ed.  pr.  15 14.  Florence,  1562. 

Guii-i-AUME  BuDic.  Born  at  Paris  in  1467  ;  died  at  I'aris,  August  23,  1540. 
Son  of  lean  Bude,  grand  audiencer  of  France.  He  became  secretary  to 
Louis  XII,  master  of  requests  to  Francois  I,  royal  librarian,  and  ambassa- 
dor to  Leo  X.  He  was  a  man  of  great  erudition,  and  was  instrumental  in 
founding  the  College  de  France. 

Title.  '  Trattato  //  delle  Monete  //  e  Valvta  loro,//  Ridotte 
dal  coftume  antico,  all'vfo  mo-//derno,  Di  M.  Guglielmo//Bvdeo. 
//Tradotto  per  M.  Giouan  Bernardo // Gualandi  Fiorentino.// 
In  Fiorenza  ;  // Apresso  I  Givnti  //  MDLXII.//  Con  licenza,  & 
Priuilegio.'     (P.  i.) 

ColopJion.  '  In  Fiorenza  apreffo  gli  heredi  di  //  Bernardo 
Giunti//i562.'    (P.  318.) 

Description.  8°,  10.5  X  17.2  cm.,  the  text  being  6.8  x  13.1 
cm.  8  pp.  unnumb.  +  3  blank  +  309  numb.  =  320  pp.,  uncut, 
28  11.     Florence,  1562. 

Editions.  The  dedicatory  epistle  is  dated  '  Da.  Viterbo  il  xxx. 
d'Agofto.  MDLXI,'  so  that  this  is  the  first  (as  it  is  the  only) 
edition  of  Gualandi's  translation  of  Budaeus.  I  have  not  com- 
pared it  with  the  '  Libri  de  asse  et  partibus  ejus,'  Paris,  15 14; 
second  edition,  Venice,  Aldus,  1522.  Although  on  the  title 
page  it  is  called  a  translation,  the  various  books,  six  in  number, 
begin  'Trattato  delle //Monete//  di  M.  Gio.  Bernardo//  Gvalandi 
Cittadino  //  Fiorentino,'  leading  to  the  belief  that  it  may  have 
been  rewritten  by  Gualandi. 

This  treatise  is  purely  historical,  describing  in  a  prolix  manner  the 
ancient  measures,  a  subject  of  interest  to  arithmeticians  in  the  sixteenth 
century  on  account  of  the  great  number  of  tables  of  denominate  numbers 
in  use  in  Italy,  France,  and  Germany. 

There  was  also  a  work  by  Budaeus  entitled  '  Minervoe  Aragonise  Assis 
Budeani  supputatio  compendiaia  ad  monetam  j^onderaque  et  mensuras 
Hispanic  nostre,'  etc.,  published  at  Saragossa  in  1536,  8°- 


loo  KARA   ARITHMETICA 

JOHANN   BOSCHENSTEYN. 

Ed.  pr.  1 5  14.  Augsburg,  15 14. 

Beschenstein,  Boeschenstain,  Bossenstein,  Boechsentein,  Buch- 
SENSTEIN,  Poschenstein,  Besentinus,  etc.  Born  at  Esslingen,  Swabia,  in 
1472;  died  in  1532.  He  taught  Hebrew  at  the  universities  of  Ingolstadt 
and  Heidelberg,  and  also  at  Antwerp  and  Nurnberg.  Luther  and  Melanch- 
thon  were  among  his  pupils. 

Title.    See  Fig.  55. 

Colophon.  '  Getruckt  in  der  Kay ferlichen  ftat  Augfpurg  durch 
//Erhart  oglin  Anno  15  14  Jar.'    (F.  24,  r.) 

Description.  4°,  14.4  X  19.2  cm.,  the  text  being  8.9  X  14-8 
cm.     24  ff.  unnumb.,  30  11.     Augsburg,  15  14. 

Editions.  Augsburg,  15 14,  4°  (here  described);  ib.,  15 16; 
ib.,  1 5 18.  Boschensteyn  is  also  said  to  have  published  at 
Auo"sburg  in  15  14  *  Ein  New  geordnet  Rechenbuchlein  auf  den 
hnien  mit  Rechenpfennigen,'  4°,  but  this  is  doubtless  Kobel's 
work  (p.  102). 

This  is  one  of  the  more  interesting  of  the  early  German  arithmetics. 
It  is  mercantile  in  character,  and  presents  in  condensed  form  the  essen- 
tials of  business  arithmetic.  Among  the  pecuUarities  of  the  book  is  the 
use  of '  figures  '  for  'species.'  Boschensteyn  gives  seven  of  these  funda- 
mental operations  :  '  Das  scind  nun  die  Siben  figuren,'  '  Die  Erft  figur 
Numeratio,'  '  Die  Ander  figur  Additio,'  etc.  He  includes  Duplatio  and 
Mediatio,  and  he  checks  all  of  his  work  by  casting  out  nines.  His  appli- 
cations are  chiefly  in  the  '  Regula  de  Try,'  partnership,  and  '  Regula 
Fufti '  (where  he  gives  his  rule  in  verse). 

JAKOB    KOBEL.     Ed.  pr.  15 14.  Augsburg,  15 14. 

Kobel,  Kobelrjs,  Kobilinus.  Born  at  Heidelberg  in  1470;  died  at 
Oppenheim,  January  31,  1533.  He  studied  at  Cracow,  where  Copernicus 
was  his  fellow-student.  He  was  a  man  of  varied  attainments,  meeting  with 
success  as  a  Rechenmeister,  printer,  engraver,  woodcarver,  poet,  and  public 
official. 

Title.    See  Fig.  56. 

Colophon.  '  Getruckt  tzii  Augfpurg  durch  Erhart  oglin.//  Anno 
M.D.XIIII.'     (F.  XXIIII,  r.) 

Description.  4°,  13.7  X  18.9  cm.,  the  text  being  9x15  cm. 
6  ff.  unnumb.  +  24  numb,  (in  Roman)  =  30  ff.,  30-35  11.  Augs- 
burg, I  5  14. 


PRINTED    BOOKS 


lOI 


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Fig.  55.   Title  page  ok  the   15 14  boschenstevn 


I02  RARA   ARITHMETICA 

Editions.  Kobel's  Rechenbuch  appeared  under  such  varied 
titles  and  in  such  different  combinations  with  his  other  books 
that  it  is  difficult  to  say  whether  a  given  edition  is  a  new  work 
or  merely  a  revision.  It  will  aid  the  student  if  he  recognizes 
in  the  first  place  that  Kobel  wrote  three  distinct  books,  (i)  the 
'  Rechenbiichlein,'  (2)  'Mit  der  Kryden,'  (3)  the  'Vysierbuch.' 
The  'Rechenbiichlein'  first  appeared  at  Augsburg  in  15 14,  4°; 
the  '  Vysierbuch,'  a  treatise  on  gauging,  at  Oppenheim,  s.  a. 
(15  I  5);  and  'Mit  der  Kryden'  at  Oppenheim  in  1520. 

When  the  1 5 1 8  edition  of  the  Rechenbiichlein  (p.  108)  appeared 
the  title  was  changed,  and  a  few  pages  were  slightly  altered. 
The  1 53 1  edition  (p.  108),  however,  shows  many  changes,  cer- 
tain chapters  being  entirely  rewritten,  and  others  considerably 
expanded.  Although  bearing  a  similar  title,  this  might  with 
some  justice  be  called  a  different  treatise  ;  and  yet  it  is  so  mani- 
festly a  revision  of  the  15  14  work  that  it  may  more  properly 
be  classed  as  a  new  edition. 

The  three  books  were  sometimes  published  as  one  and  some- 
times separately.  The  following  list  of  editions  is,  therefore, 
probably  incomplete,  and  it  should  be  understood  that  any  book 
mentioned  may  have  been  published  with  some  other  one. 

Editions  of  the  Rechenbuchlein  :  Augsburg,  1 5 14,  4°  (p.  100) ; 
Oppenheim,   1514,  4°  (P-  106);  ib.,  s.  a.  (c.  1515);  Augsburg, 
1516,4°;   1 5  17,  4°  ;  Oppenheim,  1 5  18,  4°  (p.  108  ;  '  zum  Dritte 
male  gebeffert,'  and  hence  the  third  revision,  although  at  least 
the  sixth   edition);   two  other  editions   before    1520  (Unger) 
Oppenheim,    1522,   8°;    ib.,    1525,    12°;    Frankfort,    1527,    8° 
ib.,   1531,  8°  (p.  108);   Oppenheim,  1531  ;  ib.,  1532  ;  ib.,  1535 
Frankfort,  1537,  8°  (p.  no);  ib.,  1544,  8°;  ib.,  1549;  ib.,  1564 
(p.  Ill);  ib.,  1573;   1575;  Frankfort,  1584,  8°. 

Editions  of  '  Mit  der  Kryde  od'  Schreibfedern/  durch  die  zei- 
ferzal  zu  reche  //  Ein  neiiw  Rechepiichlein/  den  angenden  Schu- 
lern  d'  rechnug  zu  ere  getruckt '  :  Oppenheim,  1520;  Frankfort, 
1537,  8°  (p.  no);  probably  included  in  various  other  editions  of 
the  Rechenbiichlein. 


PRINTED    BOOKS 


103 


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mit  TRccbcit  pfeningcn :  ocii 
3ungen3nsenocii5«bcir 

ticbem  c^cbtmcb  vno  bcno 
clnlmbtlicb5itkmcn 


Fio.  56.   Title  page  of  kobel's  Rechcnhicchlin  (15  14) 


I04 


KARA   ARITHMETICA 


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Fig.  57.    A  page  from  kobel's  Rechcnbicchlin   (1514) 


PRINTi:!)    HOOKS  105 

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2(«^  bm  obgefc^tibert  5tg««rt/tJtt  (Sicempcln  m^tg^^w 
leic^tUc^  lemert  all  br0c^/bie  biv  in  bcinat  rec^en  fiix^ 
tommcn/wic  bn  bic^  vevficen/  aftfcbrciben  mnb  anp 
fpve^en  foU/Vntf  veil  biv  rturi  ly^^cvnad)  cxbcn  /  vnnb 
^m  etlid)  :^cQ€\n/Smcn/vnnb  l^tempel  in  gcb:o^ 
^crt  naUn/xc le man  bie  bwr^  bic  3^egel  Sir?  rcc^mrt 
Dtib  rtwfp^fcrt  folic- 

frag  fnrf ombt/itt  bcren  ^)tc  cvft  $aU  gepto^ert  i(i/  t>tt5 
tic  mitttl  mb  Utfic  naU  gart^  pldbcn/  ©o  mu|Itt  bic 
ttfi  naU  an^  bvcd)m/in  ben  bnlcfe  bcrbe^  ir  scfd)vif 
bm  Pet  5Dc6  gclcid^eit  mu(iw  bie  let(^ ^al  awd)  brc*ert 
twr4?  bm  felbcn  bruc^)  vno  gcfd)i4)t  bae  auj5  b  vrfacl) 
b^cwc^ylalweg  b'fccrp  rnnb  Ictfltjale  m  bcC>ei7t»tiJ5 

gelcii^  fciit  follw/viirtb  fo  b^o  alfo  gcfct)ic^t/eo  foU 

Fig.  58.   A  page  from  kobel's  Rcchenbicchlin  (15  14) 


io6  KARA   ARITHMETICA 

Editions  of  the  '  Vysierbuch  '  :  Oppenheim,  s.  a.  (151 5).  4° 
(p.  113);  Frankfort,  1527;  ib.,  1531,  8°  (p.  108);  Oppenheim, 
1531  ;  ib.,  1532  ;  s.  1.,  1584;  probably  included  in  various  other 
editions  of  the  Rechenbiichlein  (p.  iii,  1564). 

This  is  the  first  edition  of  this  well-known  arithmetic.  As  already 
stated,  the  title  was  occasionally  changed,  but  the  work  was  essentially 
but  little  altered.  It  is  a  purely  commercial  book,  with  all  of  the  opera- 
tions performed  by  counters  as  was  still  the  custom  of  the  time  in  most 
parts  of  Germany.  Kobel  treats  of  the  rule  of  three  ('die  Gulden  Kegel, 
die  von  dem  Walen  de  Try  genant  wirt'),  partnership,  reduction,  inher- 
itances, and  exchange.  The  fundamental  operations  include  progres- 
sions, and  Roman  numerals  are  used  except  in  the  section  on  notation 
(Fig.  57).  Kobel  makes  a  curious  use  of  the  Arabic  method  of  writing 
fractions,  the  terms  being  written  in  Roman,  as  in  the  case  of 

"'  for     ^     (see  Fig.  58). 


Illic  .LX  460 

Altogether,  Kobel  was  a  vigorous  writer,  and  his  Gothic  style  shows  him 
to  have  been  no  more  a  follower  of  the  Italian  arithmeticians  than  Diirer 
and  Holbein  were  of  the  Italian  artists. 

Other  works  0/1514.  Boethius,  p.  27,  1488  ;  Grammateus,  p.  123  ; 
Jordanus,  p.  65,  1496  ;  Stromer,  p.  83,  1504  ;  Torrentini,  p.  76,  1501  ; 
Hermannus  Buschius, '  Algorithmus  linealis  Proiectiliu :  de  Ttegris  perpul- 
chris  Arithmetice  artis  regulis  :  earundemque  probationibus  claris  exor- 
natus,'  Vienna,  4°,  4  ff.  (also  catalogued  as  anonymous,  and  as  written 
by  Johannes  Cusanus ;  see  p.  43,  c.  1490,  for  Nicolaus  Cusa;  see  also 
p.  86,  1505).  There  was  also  pubhshed  c.  15 14  an  anonymous  work 
entitled  '  Arithmetics  practicae  Tractatus  qui  dicitur  Algorismus,  cum 
additionibus  utiliter  adjunctis,'  Paris,  4°. 

JAKOB   KOBEL.     Ed.  pr.  15 14.  Oppenheim,  15 14. 

See  p.  100. 

Title.  The  title  page  is  missing.  Pol.  A  ij  begins  :  '  Dem 
Ernueften  Dietherichen//Remerer  von  Wormbs:genant  von  Dal- 
burgk  :  //  meinen  befundern  gunftigen  lieben  Junck-//herzen/ 
Eubeutich  Jacob  Kobel/ //dielTer  zeit  Statfchreiber  zu  Op-// 
penheim/mein  willig  dinft // allerzeyt  bereydt//zu  vor.' 

Colophon.  '  Getriickt  zii  Oppenheym.//  Anno.  2  c  M.CCCCC. 
XIIIL'     (F.  no.  XXIIII.     See  Fig.  59.) 


PRINTED    BOOKS  107 

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feumbceffclregttt)c(fcnb5bcrui  fcynefccfcl  ffc-^abr 
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(Setiiickt  511 0ppenbcvm. 
Unito^iclliXCCCCXrai. 

Fig.  59.    Last  page  of  kobel's  KcchoibifihUn   (ofpenheim,  15 14) 


io8  RARA   ARITHMETICA 

Description.  4°,  14.  i  X  19.6  cm.,  the  text  being  9.2  X  15.8  cm. 
5  ff.  mimmib.  +  24  mimb.  =  29  ff.,  28-34  11.    Oppenheim,  15  14. 

See  p.  106.  It  is  a  curious  fact  that  the  first  and  second  editions 
should  have  appeared  in  the  same  year  at  two  different  places. 

JAKOB    KOBEL.     Ed.  pr.  15 14.  Oppenheim,  15 18. 

See  p.  100. 

Title.    See  Fig.  60. 

Colophon.  'Alfo  Endet  fich  Seliglich  d'  Drit// Truck  dif3 
Rechenbuchleins/  Zu  Oppenheym  zufam-//men  gefegt/  vnd 
volendt  vff  Dinftag  des  Hey-//ligen  Criig  Erhohungs  tag.// 
Annodni  .1518.'     (F.  XLVI,  r.) 

Description.  4°,  13.7  X  18.4  cm.,  the  text  being  9.2  X  14.8 
cm.  4  ff.  unnumb.  +46  mmib.  (I  to  XLVI)  =  50  ff.,  32-33  11. 
Oppenheim,  15 18. 

See  p.  106. 

JAKOB   KOBEL.     Ed.  pr.  15 14.  Frankfort,  1531. 

See  p.  100. 

Title.  '  Ein  new  geor-//denet  Kunftlich  Rechenbuchlin/ 
Ja-//cob  Kobels/  Stattfchreiber  zii  Oppen-//heim/  Auff  den 
Linien  vnd  Spacien/  mit  Rech-//enpfeiiingen.  Den  angehn- 
den  Schulern  Rech//nens  gantz  leichtlich  zu  lernen.  Vnd  zu 
Kellerei//en  Ampten/  Kauffmanfchafften/  vnd  Krame-//reien 
dienlich  vnd  brauchlich.  Mit  vilen  erkla-//rungen/  Leren/ 
Regeln  vnd  Exempln.// Ji@^  Mehr  dafi  vormals  ie  getruckt/ 
//gebeffert/ vnd  ziigefetzt.// Im  Jare  M.  D.  XXXL//Vifir 
Buchlin/  //Den  Jungen/  angehnden/  //  Leijfchen  Vifirern  gantz 
leichtlich  zu  lernen/  //verftehn  vnd  Rechnen.  Wie  mari  ein  // 
Vifirrut  machen/vnnd  damit  ein//iedes  vafz  vifirenn/ folle  Zii 
//ende  difes  Rechenbi!ich//lins  angehenckt.'     (F.  i,  r.) 

Colophon.  'CGetruckt  zii  Franckenfurt/am  Mein.  In  Ver// 
lag  vnnd  Gemeynfchafft  des  Ernhaffetnn  //  vnd  Fiirnemen  Herrn 
Jacoben  K6-//bels/  Stattfchreiber  zu  Oppen-//heim.  Bei  Chrif- 
tian  Ege-//nolffen.  Im  Mertzen/  //  Nach  der  geburt  //  Chrifti. 
//M.D.XXXI.  Jar.'     (F.  CXII,  r.) 


PRINTED   BOOKS 


109 


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iD46  er  rttf  arbdt  r^cr  t>mb  fflfl 
pclftampr/t>n^  :^cllcrK/t?0nc^r/ 
jiDae  cf  ntt  tt^iirb/pcrad)  t  );>a;rticl^  t/ 
Dec  ^au(f  Mg>5ucb  vii  leg  mit  petgj 
v^n^  nurcf  /  wae  tcl}  yn  ^)n^ectt)ct(5/ 
IDcm  f  onv  cr  iiocb  n)tr  firdcm  mfit 
*25c^urpn6ffot/Sccl/Cnb/t>tt  gwr* 


Fig.  60,   Title  pauk  of  kobel's  Rcchi-puchlcin   (15  18). 


no  RARA   ARITHMETICA 

Description.   8°,  9.7  X  14.9  cm.,  the  text  being  6.9  X  1 1.5  cm. 
112  ff.  numb,  in  Roman,  28-29  11.     Frankfort,  1531. 
Editions.    See  p.  102. 

It  is  often  stated  that  the  '  Jacob's  staff'  used  by  surveyors  was  first 
described  by  Kobel  in  this  year  (1531)  and  that  it  received  its  name  in 
his  honor.  The  name  was  old  before  this  time,  however,  as  applied  to 
some  form  of  surveying  instrument,  for  in  the  Margarita  Philosophica 
(1503,  Bk.  VI,  tract.  II)  is  a  conversation  between  a  master  and  his 
disciple,  in  which  it  is  mentioned  :  '  Mag.  baculo  que  lacob  dicunt. 
Dis.  Qualis  is  eft  baculus?'  The  master  thereupon  describes  the  instru- 
ment, and  a  picture  of  it  is  given. 

JAKOB    KOBEL.     Ed.  pr.  1514.  Frankfort,  1537. 

See  p.  100. 

Title.  '  Zwey  rech-//enbuchlin:  vff  der  //  Linien  vnd  Zipher/ 
Mit  eym  angehenck-//ten  Vifirbuch/fo  verftendtlich  fijr//geben/ 
das  iedem  hieraufz  on//eih  lerer  wol  zulernen.// CDurch  den 
Achtbarn  vnd  wol  erfarnen//H.  Jacoben  Kobel  Statfchreiber 
//zu  Oppenheym.//    Franc.    Chrift.  Egen.'     (F.  i,  r.) 

Colophon.   '  Ende/Im  lar  M.D.XXXVIL'     (F.  numb.  144,  r.) 

Description.  8°,  9.5  X  15  cm.,  the  text  being  7.1  X  11.9  cm. 
8  ff.  unnumb.  +  9-144  numb.  =  144  ff.,  30  11.     Frankfort,  1537. 

Editions.    See  p.  102. 

This  is  the  earliest  of  Mr.  Plimpton's  copies  containing  the  three 
books  written  by  Kobel,  (i)  '  Rechenbiichlein,'  (2)  'Mit  der  Kryde,' 
(3)  the  '  Vysierbuch.'  The  combination  of  the  three  in  a  single  volume 
formed  one  of  the  best  books  of  the  time,  giving  the  operations  both 
with  counters  and  according  to  algorism.  The  latter  is  given  in  the  part 
entitled  'Mit  der  kreiden // odder  fchreibfederen/durch //die  ziffer- 
zal  zu  rechen/  Ein  new  Rechen-//buchlin/  den  angehnden  fchulern 
der //rechnung  zu  eren  getruckt.'  (F.  106.)  In  this  work  Kobel  also 
includes  the  usual  business  problems  of  the  period  and  the  chapter  on 
the  calendar  required  by  the  Church  schools.  An  unusually  complete 
treatment  of  gauging  is  given  in  the  part  entitled  '  Eyn  new  Vifir  // 
Buchlin/  den  Leyen/  zu  //  leichtem  vnd  begreiflichem  verftandt  // 
verordnet/ Durch  H.  Jacob //Kobel  Stattfchreiber  zu //Oppenheym.' 
(F.  95,  r.)  It  is  much  more  complete  than  the  15 14  edition  of  the 
'  Rechenbiichlein  '   (see  p.  106),  and  is  substantially  the  same  as  the 


PRINTED    BOOKS  iii 

1531   edition  (see  p.  108)  exce])t  that  it  contains  the  thinl  part,  '  Mit 
der  Kreiden,'  which  the  latter  does  not. 

The  Hindu-Arabic  numerals  were  still  considered  difficult  ('  den  die 
Zifer  zal  am  erften  zulernen  fchwere,'  f.  9,  v.),  and  teachers  still  felt  it 
better  to  begin  with  the  common  Roman  characters  ('  wil  ich  ziim  erften 
die  felb  Teutfche  zal  .  .  .  hie  anzeygen  vii  erkleren  '). 


JAKOB   KOBEL.     Ed. 

See  p.  100. 

Title.  '  Rechenbiich/ 
//Auff  Linien  vnd  Ziffern. 
//MiteinemVifirbuchlin/ 
Klar  //  vnd  verftendtlich 
furgeben.  //  Gerechnet 
Buchlin/auffalle//Wahr 
vnd  Kauffmanfchafft  / 
Muntz/  //  Gewicht/  Elen/ 
vnd  Mafz/ viler  Land  // 
vnd  Stett  verglichen.// 
Durch  H.  Jacob  Kobel.  // 
(Woodcut.)  Cum  Gratia  & 
Priuilegio.//  Franckfurt/ 
Bei  Chr.  Egen.  Erben  // 
M.  D.  LXIIIL'    (F.  1,1.) 

Colop]io7i.  '  Getruckt  zu 
Franckfurt  //  am  Mayn 
bey  Chriftian  E-//genolffs 
Erben.// M.  D.  LXIIIL' 
(F.  194,  V.) 

Description.  8°,  9.  i  x 
15.2  cm.,  the  text  being 
6.3  X  12.2  cm.  12  ff. 
unnumb.  +  194  numb.  = 
206  ff.,  28-29  11.  Frank- 
fort, 1564.  (There  were 
which  this  is  the  sixth.) 


pr.   I  5  14. 


Frankfort,  1564. 


Vot7Vccf£i-t<Mfrflgcrt.  57 

cdjt fjftfl/ vtino  jl;m rtlfo  nad?Hompfl/n)tt« 
(lit  tad;tl!Vt)  (eriteit  tcdjitjc»t/  aWm  Ccs  X>\x\ti 

met  m  (Jrtnfjctt  jclen. 


^iit^ct:n>oDcc  -f^juiiffmitttet  gebct  onff 
XiiXi  mard't/  fcuffr vbetbftupt em !R,<!rbl(it 
ttittKcbnctbym/Ortrnmb  cjibt  fie  ocl?^cbcil 
pfmninj;/  fofichcun  tempt /fmOct fie itn 
Eorblitt  bunOa-t  vn&  <id;t5igbyrn/:3(tt)te 
fc«g/wtevtlbyunfievmbe(npfennfngb<J«' 
bcf  Cbit/ftfflob^dcrt/  fo  foHiptortjcbctt/ 

2(lfo  xnl  byi'cn  bn  t  fie  vmb  emcn  pfeti* 

nms  /  ViJt)  ijl  ttjolfcyl 

t)t»inb. 

iEtn2^6cl;cittJ:cy(Tcf  0el;trtrt  ctneiii  Stii* 

Fig.  61.  From  kobf.l's  Rcc)icnbuch(\^(i^ 
at  least  eight  Frankfort  editions,   of 


112  RARA   ARITHMETICA 

Editions.    See  p.  102. 

The  first  165  ff.  are  the  same  as  in  the  1537  edition  (see  p.  1 10).  The 
rest  of  the  book  is  not  found  in  the  1537  edition,  or  in  any  other 
edition  in  the  Plimpton  hbrary.  It  consists  of  a  description  of  foreign 
and  domestic  money,  with  numerous  iUustrations  of  coins.    This  begins 


nY0m^i»c5ct  m^  (l>ppcnbeym/(6t«ce 
^€»6it  ■Scyn«d)/&«  ArtOcc  ConQ  v5  ^tc- 
bctgcttftnt/woltcttmi't  crtt4nt)ei:  gen  Kom 

ctttcnMgnid?t  mcl?«^  t)cnrt?cb«ttmetk»gc^ 
|jcn/2tbec  CtJtt^  vofi  Crcbcr  voas,  mttg  vi«> 
J^crdf/©cnttdd?tdtten  wg  iMnctlcrtgcljctt/ 
5Dtf^b<JlbCttgtcng©oit  -^cyjincl?  ncun  teg 
cbc  rttti?  (Dppciibcytit  fCJitt  (Tontj  von  Ctc- 
l>e»:/2iirottJrti:eo«  ^cynnd?  €on9Ctt9o. 
tJtctIertf&ig(t«gci»/d;cConeau3cI?<?b6b<^ 
«tt^3rttgebe«»« 

nuttiftDfefrfig  /M  tt?ie  vil  t<tgcti  Cdti^ 
ijdtfCrcbctSon-^cytiticl^cttvbergangcii/ 
»nO  l>ic  Jwett  juftmcn  f  otnmcn  femt)? 

9DiCli>nt>  t)crglctd?cn  ffftgnjjtjlu  alfo  Cnfc 
fc^ct^ctt.3^um  crftctt  leg  Die  »tictlcn/Dic  6ott 
•^eynticb  £ontjcnfurgrtngcttt|l/Drts  ftmiJ 
9o.meilcrt/vffDcm^c<1)ci5bi^n<f/'X''nMf  rit 
&<?ttn;>icmc(lcn/fic£cnitucinc»wgiiKb«; 

Fig.  62.    From  K()bel's  Rechcnbuch  (1564) 

(f.  165,  r.):  'Von  Frembden  vnd  Hie//landifchen  Mflntzen/ So difer zeit 
in//Teutfch  vnnd  Welfchen  landen/inn  aller // Kauffmanfchafift  vnd 
Gewerb/  Handehi/  //viler  Land  art  im  branch/  geng/  gibig  oder  //  ver- 
ruffen  Muntzen/  .  .  .'  The  book  closes  (f.  184,  v.)  with  a  set  of  tables 
and  (f.  192,  V.)  a  '  Regifter.' 


PRINTED    HOOKS 


H3 


JAKOB    KOBEL.     Ed.  pr.  1515.         Opponheim,  s.  a.  (i  5  1  5). 

See  p.  100. 

Title.    '  Eyn  New  geordet // Vyfirbiich.  licit  yn.//Wie  man 
vff  eins  yden  Lands  Eych  // vii  Mafz/ein  gcrecht  Vyfirut  machc 


Y\o.  63.    Last  page  of  kobkl's    \'ysirhuch   (15 15) 

//vn  do  mit  ein  ygklich  onbekant  Vafz // vyfieren/ auch  feynen 
inhalt  crlernen  //  folle.      Den  anhebenden  Schvilern   ViZ/fircns 


114  RARA   ARITHMETICA 

Leichtlich/ mit  Figuren  vnnd//Exempeln/zu  lernen/angezeigt. 
//Angehengt  Tafeln.//Die  Erften  Fyer  halten  yn  gerechet/ 
11  io  eyn  Fuder  weins  kaufft  wirt/ vmb // Guide  zu  XXVI.  Od' 
XXIIII.  ATb.  ;c.//  Was  die  Ome/das  Fyrtel/  vnnd  die  Mafz 
gelten.//Die  Andern  Tafeln/Zeyge  an/ Ver//anderug  vii  wech- 
felug  einer  Muntz//durch  die  ander/als  ^9.  in  lb.  jc. //Ge- 
druckt  zu  Oppenheym.'     (F.  i,  r.) 

Colophon.  'Gedruckt  zu  Oppenheym.'    (F.  32,  v.   See  Fig.  63.) 

Description.  4°,  14.4  X  19.7  cm.,  the  text  being  9.4  X  i4-5 
cm.  4  ff.  unnumb.  +  28  numb.  =  32  ff.,  30-32  11.  Oppenheim, 
s.  a.  (151  5).  There  is  no  date  on  the  title  page  nor  in  the  colo- 
phon, but  the  prefatory  statement  closes  with  the  words,  '  Vol- 
nendet  vff  dornftag  noch  Letare.     Anno  &  c.  15  15.' 

Editions.    See  p.  106. 

The  work  is  semiarithmetical,  quite  as  much  so  as  the  chapters  on 
mensuration  in  our  textbooks ;  chapters,  indeed,  which  owe  their  origin 
in  no  small  degree  to  these  treatises  on  gauging  so  often  appended  to 
the  old  arithmetics.  This  work  is  illustrated  with  quaint  woodcuts  show- 
ing the  use  of  the  '  Vylirftab  '  or  gauging  measure.     (See  Fig.  63.) 

GIROLAMO  AND  GIANNANTONIO   TAGLIENTE. 

Ed.  pr.  15  15.  s.  1.  (Venice),  1525. 

Venetian  arithmeticians  of  c.  1 500. 

Title.  '  Opera  che  //  insegna  //  A  fare  ogni  Ragione  //  de  Mer- 
catia  //  et  a  pertegare  le  Terre  //  Con  arte  giometrical  //  Intito- 
lata  Componimeto  //  di  arithmetica  //  Con  grati"  &  preuilegio  // 
M.  D.  XXV.'     (F.  I,  r.) 

Description.  8°,  10.3  X  15.1  cm.,  the  text  being  7.5  X  12.5 
cm.     91  ff.,  28-33  11-     S.  1.  (Venice),  1525. 

Editions.  Venice,  15  15,  8°  (De  Morgan  having  erred  in  say- 
ing'apparently  before  1500');  ib.,  1520;  ib.,  1523,8°;  ib.  (s.  1.), 
1525,  8°  (here  described);  ib.,  1526;  ib.,  1527;  ib.,  1528,  4°; 
s.  a.  (1530.?),  8°;  Venice,  1541  ;  Milan,  1541,  8°  (p.  115);  s.  1., 
1547,  8°;  Milan,  1548;  Venice,  1548  (with  probably  a  second 
Venetian  edition,  i  548,  under  the  title  '  Thesoro  universale  de 
abacho,'  by  '  lucha  atonio  de  Uberti,'  8°);ib.,  1550;  ib.,  1554;  ib., 


PRINTED    BOOKS  115 

1557;  ib.,  1561;  ib.,  1564;  ib.,  1567;  ib.,  1570;  Milan,  1570; 
ib.,  1576;  ib.,  1579;  ib.,  1586.  Riccardi  also  mentions  eleven 
other  editions,  s.a.,  and  four  such  appeared  in  the  Boncompagni 
sale,  and  four  in  the  Fisher  sale  of  1906.  The  work  also  ap- 
peared, and  is  frequently  catalogued,  without  the  authors'  names. 

These  various  editions  have  been  the  object  of  critical  study 
by  E.  A.  Cicogna  and  Prince  Boncompagni.  The  former  set 
forth  his  results  in  his  Saggio  di  bibliogmfia  I \ncziana,  Venice, 
1847,  p.  218,  ascribing  the  work  to  Girolamo  Tagliente  'con 
I'ajuto  del  suo  consanguineo  Giannantonio  Tagliente.'  In  the 
edition  of  1525,  here  described,  only  the  former  name  appears, 
the  text  beginning,  'CAl  benigno  lettore//Hieronymo  Tagliente.' 
Boncompagni's  investigations,  setting  forth  the  differences  in  the 
various  editions,  appeared  in  the  Atti  dell  Accadcinia  Pontif.  de 
Nuovi  Lificei,  XVI,  139,  147,  155,  304.  See  also  Riccardi,  I, 
2,  484,  and  Boncompagni's  Bulletino,  XIII,  247. 

There  was  also  a  treatise  published  by  the  Taglientes  entitled 
'  Regole  di  mercatura  intitolato  componimento  di  arithmetica,' 
Venice,  1524,  8°,  probably  another  edition  of  this  work.  See 
also  the  treatise  on  bookkeeping  mentioned  on  p.  141,  1525. 

The  book  opens  with  a  brief  treatment  of  notation  and  finger  symbols. 
Then  follow  in  order  the  multiplication  table,  the  proof  of  sevens,  various 
methods  of  multiplication,  division  by  the  galley  method,  addition  chiefly 
of  denominate  numbers,  subtraction,  the  operations  with  fractions  in  the 
same  order,  exchange,  rule  of  three,  and  applied  problems.  There  are 
numerous  interesting  woodcuts,  and  such  familiar  problems  as  those  of 
the  couriers,  the  testament,  and  the  sale  of  eggs  are  given  with  illustra- 
tions. In  spite  of  the  arrangement  of  topics,  there  were  few  textbooks 
so  influential  as  this  in  shaping  the  subsequent  teaching  of  arithmetic. 

GIROLAMO  .'^ND  GIANNANTONIO   TAGLIENTE. 

Ed.  pr.  15  1 5.  Milan,  1541. 

See  p.  114. 

Title.  '  Libro  //  dabaco  che  in//segna  a  fare  //  ogni  ragione 
mercadantile,  &  //  pertegare  le  terre  co  I'arte  di  //la  Geometria, 
e  altre  no//bilifsime  raginoe  ftra-//ordinarie  c5  la  Ta-//riffa  come 


ii6  RARA   ARITHMETICA 

refpon//deno  li  pefi  &  //  Monede  de  moltc  terre  del  mon-/7do  con 
la  inclita  citta  di  Vene-//gia.  Elquel  Libro  fe  chiama  //Thefauro 
vniuerfale.'     (F.  i,  r.) 

Colophon.  '  Stampato  in  Milano  per  lo.  Antonio  da  Borgho.// 
Nell' anno  del.  M.  D.  XLL'     (F.  80,  v.) 

Description.  8°,  10  X  14.6  cm.,  the  text  being  j .J  X  12.3  cm. 
80  ff.  unnumb.  +  7  blank  =  87  ff.,  23-30  11.     Milan,  1541. 

Editions.    Seep.  114. 


Fig.  64.    From  the   1541  tagliente 

This  differs  but  little  from  the  1525  edition,  the  ' Opera  che  insegna' 
(see  p.  114),  except  in  having  a  set  of  tariff  tables  at  the  end:  '  CQui 
comenza  el  terzo  Libro  di  la  fruttifera  opera // chiamata  la  Tariffa ' 
(f.  K  iiii,  v.).  There  is  some  slight  change  in  the  phraseology,  particu- 
larly at  the  beginning  of  the  various  sections.  For  two  curious  forms  of 
multiplication  see  Fig.  64.  Such  arrangements  of  the  work  in  multipli- 
cation were  quite  common,  particularly  in  the  early  Spanish  and  Italian 
arithmetics  of  the  first  half  of  the  sixteenth  century.  That  they  should 
have  found  place  in  a  popular  mercantile  treatise  is,  however,  rather 
surprising. 


PRINTED    BOOKS  117 

VARIOUS   AUTHORS.     Ed.  pr.  1515.  Vienna,  15  15. 

Joannes  de  Mukis  (Jean  de  Metirs,  Murs,  Mi;ria)  was  born  in 
Normandy,  c.  1310;  died  after  1360.  He  wrote  on  arithmetic,  a.stronomy, 
and  music. 

Thomas  Bradwardin.    See  p.  61. 

NicoLAUs  IIorem  (Nicolas  Oresme)  was  born  at  Caen  (.')  c.  1323; 
died  at  Lisieux,  July  11,  13S2.  He  taught  in  the  College  de  Navarre  at 
Paris,  and  in  1377  became  Bishop  of  Lisieux.  He  wrote  also  an  '  Algoris- 
mus  Proportionum,'  in  which  the  idea  of  fractional  exponents  first  appears. 

Georg  von  Peurbach.    See  p.  53. 

Joannes  de  Gmunden  (Johann  von  Gmunden,  Johann  Wissbier? 
NvDEN  ?  ScHiNDEi-.'  JoHANNES  DE  Gamundia)  was  born  c.  13S0,  at 
Gmunden  on  the  Traunsee,  or  Gemiind  in  Lower  Austria,  or  Gemiind  in 
Swabia;  died  at  Vienna,  February  23,  1442.  He  was  educated  at  Vienna, 
and  taught  there,  being  the  first  professor  of  mathematics  alone  in  Austria. 

Title.    See  Fig.  65. 

Colophon.  '  Impreffum  Vienne  per  Joannem  Singrenium  // 
Expenfis  vero  Leonardi  :  Luce  Alantfe //fratrum  Anno  domini 
.M.ccccc.xv.// Decimono  die  Maij.'     (F".  54,  r.) 

Description.  4°,  14.3  x  18.4  cm.,  the  text  being  9.7  x  15.5 
cm.  (without  the  marginal  references).  54  ff.  unnumb.,  26  11. 
Vienna,  15  15. 

Editions.  There  is  no  other  edition  of  this  combination  of 
works.     See  pp.  53,  61,  118,  for  the  individual  treatises. 

This  interesting  work  consists  of  five  parts.  The  first  is  the  arith- 
metic of  Joannes  de  Muris.  While  it  is  called  an  extract  from  the 
arithmetic  of  Boethius,  it  is  merely  suggested  by  that  treatise  and  is 
really  the  work  of  Muris.  This  part  of  the  work  begins  on  f.  2  with 
the  following  title:  '  Incipit  Arithmetica  comunis  ex//diui  Seuerini 
Boetij  Arithmetica  per  M.  Joannem  //  de  muris  compendiofe  excerpta. 
//  Prohemium.'  De  Morgan  (p.  3)  mentions  a  possibly  earlier  edition, 
s.  1.  a.,  4°.     (See  also  p.  86,  1505.) 

The  second  part  of  the  volume  begins  on  f.  17,  v.,  and  is  the  work 
on  proportion  by  Thomas  Bradwardin.  It  is  a  theoretical  treatment  of 
the  subject,  and  has  the  following  title  :  '  Tractatus  breuis  proportionu  : 
ab-//breuiatus  ex  libro  de  Proportionibus.  D.  Thome  //  Braguardini 
Anglici.' 

The  third  part  begins  on  f.  27,  v.,  and  is  a  treatise  by  Nicolaus 
Horem,  with  the  following  title  :  'Tractatus  de  Latitudinibus  forma-// 
rum  fcdm  doctrina  magiftri  Nicolai  Horem.'  This  subject  attracted 
considerable  attention  in  the  latter  part  of  the  Middle  .Ages. 


ii8  RARA   ARITHMETICA 

The  fourth  part  is  the  algorism  of  Peurbach,  and  has  the  following 
title  on  f.  37,  V. :  '  Opufculu  Magiftri  Georgij  //Peurbachij  doctifs.'  As 
already  stated  (p.  53),  this  work  takes  up  the  four  fundamental  opera- 
tions and  progressions,  giving  merely  a  theoretical  discussion  of  the 
subject. 

The  fifth  part  (f.  44,  v.)  begins  :  '  Incipit  tractatus  de  Minucijs 
phi-//ficis  :  compofitus  Vienne  Auftrie  per  magiftrum //Joannem  de 
Gmunden.'  This  is  the  treatise  of  Gmunden  on  sexagesimal  fractions, 
or,  as  they  were  called  in  the  Middle  Ages,  physical  fractions.  These 
fractions,  still  used  by  us  in  our  degrees  (or  hours),  minutes,  and  seconds, 
served  the  purposes  of  the  later  decimal  fractions.  They  were  carried 
much  farther  than  is  now  the  case,  a  number  like  3°  15'  40"  15'"  45 'V 

meaning  merely  3  +  ^  +  g  +  ^3  +  ^. '  or  3  ^^  •    The  present 

symbolism  (°,  ',  ")  is  relatively  modern. 

The  book  is  particularly  interesting  because  it  combines  in  one 
volume    five   well-known   books    by  mediaeval  writers.     In   no  other 

iContmutnhocliMlo. 

^ritbmctia  commuin'o* 

•pzopojtioiiee  b:cnc0, 

fbchtimmibm  foztmrmn. 

3lgo:itbmu0.0>.0co:gij  "^curbacby  in  iimme. 

aiGo^itbniu00>a^ifln'3oaiini0Oci5munocii 
Dcmiimcgo  pbtficio. 

Fig.  65.    Title  page  of  the  Vienna  work  of  15 15 

single  volume  could  the  inadequacy  of  the  mediaeval  treatment  of 
mathematics  be  better  seen.  Indeed,  a  manuscript  of  15  15,  found  by 
Gerhardt  in  the  Wolfenbiittler  Bibliothek,  expressly  states  that  the 
lectures  on  arithmetic  given  in  the  universities  of  that  period  were 
based  on  the  above  works  of  Muris,  Bradwardin,  Peurbach,  and  Joannes 
de  Gmunden.  i^Monatsherichte  der  K.  P.  Akad.  d.  Wisscnsch.  zu  Berlin, 
1867,  p.  43.) 


PRINTED    BOOKS  119 

JOANNES    DE   MURIS.     Ed.  pr.  1515.  Mam/,  153.S. 

See  p.  117. 

Title.  '  Arilhnic//ticac  specvlativae // Libri  duo  loannis  de 
Muris  ab  in-//numeris  erroribus  quibus  hactc-//nus  corrupti,  & 
uetuftate  fer//me  perierant  diligen-//ter  emendati,//  Pvlcherri- 
mis  qvoqve  exemplis,  Formisq;  nouis  declarati  &  in  //  ufum 
ftudiofae  iuuentutis  Mogun-//tinae  iam  reccens  ex-Z/cufi.// Mo- 
gvntiae  cxcvdebat // Ivo  Scoffer  anno.// M.  D.  XXXVIII.' 
(F.  I,  r.) 

Colophoji.  '  Mogvntiae  cxcvdebat  //  Ivo  Schcffcr  anno.// 
M.  D.  XXlCVm.'     (F.  90,  V.) 

Description.  8°,  9.6  X  14.5  cm.,  the  text  being  d.G  X  1 1.9  cm. 
4  pp.  unnumb.  +  3-88  numb.  +  5  blank  +  i  with  woodcut  =  96 
pp.     Mainz,  1538. 

Editions.  This  is  the  second  dated  edition  of  the  arithmetic 
of  this  popular  mediaeval  teacher.  (See  p.  1 17.)  It  is  more  com- 
plete than  the  one  of  1515,  but  it  does  not,  like  the  latter,  give 
the  marginal  references  to  Boethius,  upon  which  it  is  so  largely 
based.    (See  also  Boethius,  p.  27.) 

See  p.  117. 

JOANNES    FCENISECA.    Ed.  pr.  1515.        Augsburg,  1515. 

An  Augsburg  teacher  of  c.  1500. 

Title.    See  Fig.  66. 

Colophon.  '  Impreffa  Augufte  Vindelicorum/communibus  im- 
penfis  lo/  //annis  Miller  atq5  loannis  foenifece.  Anno  a  natiuita 
//te  domini.    M.D.  XV.  ad.  IIII.  Cal.'  Maias.'    (F.  20,  r.) 

Description.  4°,  15.1  X  21  cm.,  the  text  being  11  X  13.8  cm. 
20  ff.  unnumb.,  7-39  11.     Augsburg,  15  15. 

Editions.    There  was  no  other  edition. 

This  is  an  extract  from  a  larger  volume,  for  the  folios  have  been 
numbered  by  hand  40-59,  and  the  register  begins  with  '  aa  i.'  Only 
two  pages  (aa  ii,  v.,  and  aa  iii,  r.)  are  devoted  to  '  Arithmetica,'  and  these 
relate  only  to  the  Boethian  system.  The  rest  of  the  book  is  devoted 
chiefly  to  geometric  figures,  the  mediaeval  astronomy,  and  music.    Such 


20 


RARA   ARITHMETICA 


a  book  shows  the  superficiality  and  general  emptiness  of  the  work  of 
the  schools  that  were  supposed  to  stand  for  culture  in  the  period  of  the 
early  Renaissance. 


^ 


Opera  Toannfs  Fanifccae  Augn* 
hcc  in  lehabenc. 

Quadratum  (apienri?:continens  in  fefcptem 

artcslibcraks  veterum. 

Circulos  biblj^  iiii.inquibus  mccaphyfica 

moiaica. 

Commentaria  honim. 

Ad  Hcc/libri  rubricainftrius 

f ignati :  ncccflarii  fu  nc» 


f 

F 


P^Grammadca 
fmno        Logjca 
Rhctorica 
Monaftica 
fOo5  Oeconomica 

.i.mosr^Politica 

j^  Alcajithmus  fubaltcrnxis 
Arichmecica 
Gcomctria 
mathcacica        Pcripcaiua  fiibalrana 
Mulica 
Aftronomia 
^^^Gcographia 
{ — Phylica 
philofbplua 

^  Mcdici'na  fubaltcrna 
tlicologia         Metaphyfica 


don.alcx.gua.lafca.rocrf 


nouus 

boerius 

boccius 

pcTUsiacobi 

bocruis 

boccius    €phcmo'i^« 

pcolcm^us      Hiftond 

diofcorides 
biblium  tripTcx 
aai. 


Fig.  66.   Title  page  of  F(t:niseca's   Opera  (15 15) 


ANONYMOUS.     Ed.  pr.  c.  15  15.  Leipzig,  s.  a.  (c.  15  15). 

Tzt/e.  '  Melchiar  Lotthervs  Ivnior  candido  lectori  salutem.' 
(Line  i.)  '  Articularis  Bedae  prefbyteri  numerorii  computatio.' 
(Line  24.) 

Description.  One  sheet,  28  X  36.4  cm.,  the  text  being  21.6  X 
26.7  cm.    42  11.    Printed  on  one  side  of  a  single  sheet. 


PRINTED    1U)UKS  121 

Editions.  There  was  no  other  edition,  antl  only  this  co])y  is 
known  of  this  one. 

This  broadside  was  published  by  Lotter  c.  15  15,  and  is  a  brief  state- 
ment of  the  numerical  finger-symbolism  of  the  ancients,  particularly 
as  described  by  the  \'enerable  Bede.  The  syml)olism  is  practically  the 
same  as  that  described  by  such  writers  as  Paciuolo  and  Aventinus  (see 
pp.  57  and  136.) 

CASPAR    LAX.     Ed.  pr.  15  15.  Paris,  15  15. 

Born  at  Sarinena,  Spain,  c.  1487  ;  died  at  Saragossa,  February  23, 1560.    He 
taught  at  Paris  and  Saragossa.    Ilis  only  works  are  the  two  here  described. 

Title.  See  Plate  IIP  '  Proportiones  magiftri  Gafparis // lax 
aragonenfis  de  farinyena  .  .  .'  (F.  loi,  r.  Separately  cata- 
logued, see  below.) 

Colophon.  '  Explicit  Arithmetica  fpeculatiua  Magiftri  Gafparis 
Lax  Aragonenfis  de  Sarinyena  duode-//cim  libris  demonftrata. 
Impreffa  Parifius  opera  ac  characteribus  Magiftri  Nicolai  de  la 
barre-Z/Expenfis  honefti  viri  Hemundi  le  feure  Bibliopole  Pari- 
fius in  vico  diui  Jacobi  fub  figno  Crefce//tis  albi  vitam  degentis. 
Anno  Domini  .1515.  Die  vero.i3.MenfisDecembris.'  (F.  100,  v.) 

Description.  Fob,  19.8  X  26.5  cm.,  the  text  being  1 4.1  X  21.8 
cm.     100  ff.  unnumb.,  59-61  11.     Paris,  15 15. 

Editions.    There  was  no  other  edition. 

A  very  prolix  treatment  of  theoretical  arithmetic,  based  on  Boethius 
and  his  mediaeval  successors.  As  the  title  shows.  Lax  was  a  Spanish 
teacher,  one  of  several  from  the  southern  peninsula  who  taught  in  the 
University  of  Paris  in  the  fifteenth  century.  Among  the  others  were 
Rollandus  (originally  from  Lisbon,  mentioned  later  in  connection  with 
the  manuscripts)  and  Ciruelo  (p.  58).  All  of  the  contributions  of  these 
scholars  were  of  this  general  theoretical  character.  De  Morgan  face- 
tiously remarks,  'For  anything  that  appears  the  author  (Lax)  could  not 
count  as  far  as  100.' 

GASPAR    LAX.     Ed.  pr.  1515.  Paris,  1515- 

See  above. 
Title.    '  Proportiones  magiftri  Gafparis //lax  aragonenfis  de  fa- 
rinyena.//Venundatur  Parifius  In  vico  diui //Jacobi  ab  Emundo 
le  feure  fub  fi-//gno  crefcentis  albi  vitam  degente.'    (F.   10 1,  r.) 


122  RARA   ARITHMETICA 

CoIopJion.  '  Expliciunt  proportiones  Ma//giftri  Gafparis  Lax 
Aragonen//ris  de  Sarinyena  impreffe  Pa-//rifius  opera  Magiftri 
Nicolai  //  de  la  barre  pro  Emundo  le  feure  //  Anno  dni  M.  d.  xv. 
die  vo  vi.  me/Zfis  Octobris.'     (F.  26,  126  of  the  whole  book,  r.) 

Dcscriptio7i.  Fol.,  19.5  X  26.6  cm.,  printed  in  double  columns, 
each  being  7  x  21.5  cm.     26  ff.  unnumb.,  66  11.     Paris,  15  15. 

Editions.    There  was  no  other  edition. 

Bound  with  the  '  Arithmetica  Speculativa'  (p.  121).  This  is  a  prolix 
treatment  of  mediaeval  ratios  after  the  Boethian  manner,  and  as  such  it 
ranks  with  works  like  those  of  Bradwardin  (p.  61),  Jordanus  (p.  62), 
and  Faber  Stapulensis  (p.  82). 

Other  works  of  15 15.  Boethius,  p.  27,  1488  ;  Bradwardin,  p.  61, 
1495;  Kobel,  p.  102,  1514;  Lanzut,  pp.83,  97,  1504,  1513;  Licht, 
p.  70,  1500  ;  Ortega,  p.  93,  1512;  Peurbach,  p.  53,  1492;  Torrentini, 
p.  76,  1 501  ;  Juan  Andres,  '  Sumario  breve  de  la  practica  de  la  arithme- 
tica,' Valencia  (from  the  book  it  appears  that  it  was  written  in  Saragossa 
in  15 14  ;  it  was  reprinted  at  Seville  in  1537)  ;  V.  Rodulphus  Spoletanus, 
'  De  proportione  proportionvm  dispvtatio,'  Rome,  4°. 

Works  of  1^16.  Johann  Boschensteyn,  p.  100,  15 14;  Capella,  p.  66, 
1499;  Ciruelo,  p.  60,  1495;  Kobel,  p.  102,  1514;  Stromer,  p.  83, 
1504  ;   Widman,  p.  36,  c.  1488. 

PIETRO    MARIA    BONINI.     Ed.  pr.  1517.    Florence,  1517. 

A  Florentine  writer  of  the  first  half  of  the  sixteenth  century. 

Title.  '  Lvcidario  darithmetica.'  (Large  woodcut.  F.  3,  r. 
Fig.  6t.) 

Colophon.  '  Cimpreffo  nella  excelfa  cipta  di  Firenze  per// 
Gianftephano  di  Carlo  da  Pauia//adi  7  di  Gennaio.'    (F.  18,  r.) 

Description.  8°,  10. 9  X  15.9  cm.,  the  text  being  8  X  12.7  cm. 
19  ff.  unnumb.,  24  11.     Florence,  15  17. 

Editions.  There  was  no  other  edition.  This  interesting 
volume  was  known  to  De  Morgan  only  by  hearsay  when  he 
wrote  his  Arithmetical  Books.  It  came  into  his  possession, 
however,  after  that  work  was  published,  as  is  shown  by  his 
autograph  on  the  title  page  (see  Fig.  ^J).  It  is  not  often  men- 
tioned by  bibliographers,  and  is  one  of  the  rare  books  of  the 
century.     In  the  Boncompagni  sale  (no.  144 1)  there  is  mentioned 


gwrt(I5a6)ai:iB  ^a]C -grpmro 


Pi  ATE  III.    TriLK  i'.\(;k  ok  lax 


PRINTED    nuOKS 


123 


([LVCIDARIO  DARITHMETICA. 


an  edition  of  1547;  but  this  is  a  misprint  for  i  5  17,  as  appears  from 
Riccardi,  vol.  I,  col.  153-4-  Riccardi  mentions  only  three  copies 
known  to  him  or  to  Boncompagni, 
but  there  was  one  in  the  Fisher  sale 
(London,  1906). 

The  book  is  a  small  octavo,  the  first 
two-thirds  being  given  to  mercantile  prob- 
lems on  exchange  and  the  reduction  of 
money.  The  last  part  treats  exclusively  of 
mensuration  :  '  Speculation!  geometriche 
di  piu  sorte :  &  jirima  lacpiadratura  del 
triangolo.' 

Other  works  of  1517.  Anonymous, 
'Algorithmus  linealis'  (see  Widman),  p.  36, 
c.  1488;  Borghi,  p.  16,  1484;  Feliciano, 
p.  146,  1526;  Kobel,  p.  102,  1514  ;  Reisch, 
p.  82,  1503;  Sacrobosco,  p.  32,  1488; 
Widman,  p.  36,  c.  1488. 


Fig.  67.    TriLK  pack 
BONiNi's  Lvcidario 


HENRICUS   GRAMMATEUS. 

Ed.  pr.  15  18.  Frankfort,  1535. 

Heinrich  ScHREiiiER;  Henricus  Scriptor  ;  Latinized  Greek,  Gram- 
MATEUS.  Born  at  Erfurt,  at  least  as  early  as  1496.  lie  describes  him- 
self as' HenrichGrammateuS// von  Erffurt/der  fiben  freien  kiinften  Meyfter.' 
He  was  a  student  at  Cracow  and  at  the  University  of  Vienna  {1507).  The 
dates  of  his  birth  and  death  are  unknown,  but  a  record  at  Vienna  reads : 
'  Anno  domini  millesimo  quingentesimo  septimo  in  festo  sanctorum  Tibureii 
et  Valeriani  martirum  .  .  .  Henricus  Scriptoris  de  Erfordia.'  He  also 
taught  at   Vienna. 

Title.    See  Fig.  68. 

Description.  8°,  9.2  X  15.2  cm.,  the  text  being  G.-j  ^  1 1.3  cm. 
96  ff.  unnumb.,  31  11.     Frankfort,  1535. 

Editions.  Vienna,  1518,8°;  Frankfort,  1535  (here  described) ; 
s.  1.  (Frankfort),  1544;   Frankfort,  1572. 

Grammateus  also  published  an  '  Algorithmus  proportionum 
una  cum  monochordi  generis  Dyatonici  compositione  .  .  . ', 
Cracow,  15 14,  4°;  '  Libellus  de  compositione  regularum  pro 
vasorum  mensuratione,'  Vienna,  15 18;  '  Behend  unnd  khunst- 
lich  Rechnung  nach  der  Regel  und  welhisch  practic,'  Niirnberg, 


124 


RARA   ARITHMETICA 


1 52 1,  8°,  an  extract  from  the  work  here  described  ;  '  Algorismus 
de  integris   Regula  de  tri  cum  exemplis,'    Erfurt,   1523;   '  Eyn 


^ 


C  (DemeyncnKegfln^f  tre. 


\cC'Cird3cnp:«rtic.      ^f0clnfalfr« 

<^**vs.p:opo:tionbe60errtn06/in©wtoi» 

/  0elpfciffcn/vnbanb:e3n|trijm2t/ 
I.  ^ur4^crfm^urt0Plt{)^0o:e. 
IIBu4^^Alun  ^urc&^a^  ioinal/'Bape  vn5 

ilTifterrSten  5U  mx^en  Dui'd)  ^cn  (Dtto^ 
&i«r/  vtit>  ^ri«n0c!/niit  anOcrn  lofligm 
(lurfen  ^er(0comctrei. 


Fk;.  68.    Title  page  of  the   1535  grammateus 

kurtz    newe    Rechenn    unnd    Visyrbuechleynn    gemacht   durch 
Heinricum  Schreyber,'  Erfurt,  1523. 

That  the  book  was  written  at  Vienna  in  15  18  appears  by  the  dedi- 
cation to  '  Dem  Edlen  furfichtigen  weifen  Johan//fen  Tfchertte  einer 


PRINTED    BOOKS  125 

des  Senats  zu  Wien,'  which  ends:  '(iebe//zu  Wi//en  in  O-Z/ricrrcich 
im  jar//nach  der  geburt  vn-//fers  Seligmachers.  M.  I).  XViij.'  In  the 
chapter  on  bookkeeping  is  the  date  1535,  so  that  probably  the  work 
was  revised  for  this  edition. 

Zl})it  fern  $u  abbiren  bieqUMtitet  tinte  uafi 
mcne/aleJ^.mitn.pnmarmitpnma/Tccunbd 
mitffcub«/tfrtiamittertia:c.Tnbmatib:au^ 
^et  fold^ejctdjen  ale  -j-  ifi  mt^t/vnb  ^/fnirt 
l)kv/in  roclc^cr  {ein  $0  tmvchn  b:ei  K(0eK 
|reicjJ:rf!Kf0cI. 

TOann  cin  quantitct  ^at  an  beyben  citen  ~(- 
«ber — fo  fol  man  n  folc^e  quanttret  abbirn  ^irt 
$0  gcfa^t  bae^cf^cn-^-obet  — 
old  9  P"« 


6  pzu 


-7^.    <yp:i.— 4t^. 


i«iti5pji.-f:'it^«  i4pn.— 141^. 
C^teanber^egd. 
3fl  in  b«  6bcrn  quantitet  —j-  vnb  in  bev  vn 
betn  —/  vnb  — )-  tibcrtrifft  — /  fo  fol  bie  rnbcr 
quantitet r on ber  6bcrn fubtral)  rt  weibcn/m 
$ubcmub:ictenfc^~|-@oabcrbtc  rnber  qua 
titet  if?  grSjfcr/fo  fubtra^ir  l^ie  Eldncrn  r  0  bee 
grSffern/rn  5U  bem  bae  bo  bleitenb  i(l/fcRC— 
ds   (?p:i.-l-6'N:      4p:r.-|-2N. 

i2p;L-~4N»        gp?r  — gN. 

i8p2U— |-2N.        lop2I 4N. 

C©ie&:itt  Kegel. 
60  mberobgcfanrcnqufintttet  rrtirrfunbe 
—  i>nb  in  ^cr  rM&ern  -{-/icrib  — -  ubertri  jf  1 4-/ 
ro|iibri'flbir<^"'^^'Ctnbcmanbern/v;nb$iimuj» 
b«0cnfd3:cib  — 3(?c6abcr/ba6bicrnbcrqua 
titet(ibcirn|fcb!c6bcrn/fo5fe|)cctn6vci)bem 
flnbcrn/pnb  3U  bf  m  crHen  fe^e -|- «l6 

Fic.  69.    From  the  1535   grammateus 

The  work  is  for  the  most  part  a  mercantile  arithmetic,  the  opera- 
tions being  given  according  to  both  the  abacists  (with  counters)  and 
the  algorists  (by  the  Hindu-Arabic  numerals),  and  a  chapter  on  book- 
keeping being  appended.  Grammateus  gives,  however,  some  considera- 
tion to  the  theory  of  numbers,  the  rules  of  the  Coss  (algebra),  music. 


126  RARA   ARITHMETICA 

bookkeeping,  and  gauging.  In  the  treatment  of  the  '  Regula  falsi,'  or 
rule  of  false  position,  the  signs  +  and  —  are  first  found  in  this  connec- 
tion (f.  Eiij).  (See  Widman,  p.  40,  1489,  who  uses  them  for  another 
purpose,  and  Vander  Hoecke,  p.  183,  1537.)  Grammateus  also  uses 
these  signs  in  writing  algebraic  binomials,  as  shown  in  Fig.  69.  It  is 
interesting  to  know  that  Rudolff  (p.  150)  learned  algebra  from  Gram- 
mateus, as  he  states  in  the  following  words  :  '  Ich  hab  von  meister 
Heinrichen  so  Grammateus  genennt  /  der  Cofs  anfengklichen  bericht 
emphangen.    Sag  im  darumb  danck.' 

Other  works  0/1518.  Boschensteyn,  p,  100,  1514  ;  Calandri,  p.  48, 
1491;  Fehciano,  p.  145,  1526;  Kobel,  p.  102,  1514;  Riese,  p.  139, 
1522  ;  Torrentini,  p.  76,  1501  ;  P^rez  de  Oliva,  '  Dialogus  in  laudem 
Arithmeticse,'  Paris. 

ANONYMOUS.     Ed.  pr.  15  19.  Venice,  15 19. 

Title.    See  Fig.  70. 

Colophon.  'Venetijs  in  Edibus  Petri  Liechtenftein // Anno 
virginei  partus  .  15  19.'    (¥ .  12,  v.) 

Description.  4°,  15.6  X  20.4  cm.,  the  text  being  10. i  X  16.3 
cm.     12  ff.  (I  blank),  33-38  11.     Venice,  15  19. 

CopfttU0  nonu67ecdefia(!iCU8:to«u0 

fo-cSiftrononijefundamcntumpulcbemmunicoii^ 

tincfis.  /Clerico  non  minus  vtilis  $  neccf/ 

farms:  cum  additionibus  quibuf/ 

dainnouitcrapzelfis* 

SSnno.  1519  •^cnett)6tn  iSdibus 
pcrri  jUiccbtcnftcin* 

Fig.  70.   Title  page  of  the   Copiitiis  notius 

Editions.  There  was  no  other  edition,  so  far  as  I  know, 
although  there  are  several  anonymous  computi,  and  some  may 
be  the  same  as  this. 

This  is  one  of  the  rare  works  setting  forth  the  computus  as  it  was 
taught  in  the  Church  schools  of  the  Middle  Ages.    (See  p.  7.) 


PRINTED    HOOKS  127 

The  verses  on  the  calendar  mentioned  on  p.  ^^  (Anianns,  1488) 
here  appear  as  follows  : 

'Aprilis:  Junius:  September  atq^  Nouember 
Hij  trij^enta  habent  vnum  reliqui  fuperaddunt 
Februarius  vigint  infuper  octoqj  dies.'     (F.  9.) 

0//ier  luorks  of  i^^ig.  Feliciano,  p.  146,  1526  ;  Blasius,  j).  97,  15  14  J 
Widman,  p.  37,  1489. 

JOHANNES   FRANCISCUS   PICUS   MIRANDULA. 

Ed.  pr.  1520.  S.  1.,  1520. 

A  nephew  of  Pico  da   Mirandola,  and  biographer  of  his  uncle.     lie  was 
murdered  in  1 533.     Like  his  uncle  he  was  a  savant  of  reputation. 

Title.  '  loannis  Francisci  Pici  Mirandvlae  domini,  et  //  Con- 
cordiae  comitis,  examen  vanitatis  do//ctrinae  gentivm,  et  veri- 
tatis  Chri-//stianae  disciplinae,//distinctvm  in  libros  sex,  qvorvm 
tres  //  omnem  philosophorvm  sectam  vni-//versim,  reliqvi  Aris- 
toteleam  //et  Aristoteleis  armis  //particvlatim  im-//pvgnant.// 
vbicvnqve  avtem  Christiana  et  //  asseritvr  et  celebratvr  //  dis- 
ciplina.'     (F.  i,  r.) 

Description.  Fol,  20.6  X  30.6  cm.,  the  text  being  16.9  X  24  cm. 
6ff.  unnumb.  +  208  numb.  +  i  blank  =  215  ff.,  44  11.    S.  1.,  1520. 

Editiojis.  There  was  no  other  edition.  The  dedication  (f.  2,  v.) 
bears  date  M.  D.  XX,  and  the  privilege  M.  D.  XIX. 

The  book  hardly  deserves  place  in  a  list  of  this  kind.  It  has,  how- 
ever, been  included  because  of  the  following  brief  chapters  on  the 
nature  of  arithmetic  :  '  Quod  fuper  mathematicis  artibus  arithmetica 
&  geometria,  fuperq^  mediis  Aftrologia  &  mufica,  gentium  philofophi 
non  conueniunt.  Cap.  vii '  ;  '  De  opinione  pythagoricoru,  &  de  ratione 
et  philolai  &  poftidonii  .  .  .  Cap.  ix '  ;  '  Quid  aduerfus  arithmetica 
facultate  pyrrhonii  difputauerint.    Cap.  vii '  (of  liber  III). 

ANONYMOUS.  S.  1.  et  a.  (c.  1520). 

Title.  'Von  dem  Rachnen  atiff  den  Linien.'  (Running 
headline.) 

Thirteen  fragments  of  proof  sheets  of  an  unknown  Cierman  arith- 
metic, three  duplicates.  The  date  is  purely  conjectural.  The  work  was 
of  at  least  46  pages,  since  the  folios  were  numbered  and  part  of  f.  23 


128  RARA   ARITHMETICA 

is  among  the  fragments.  The  title  of  the  book  probably  appears  in  the 
running  headline  above  given,  although  this  may  be  the  title  of  only 
part  of  the  work.  There  was  at  least  one  illustration  of  line  reckoning 
in  the  book.  The  lines  are  6.4  cm.  in  length,  but  there  is  no  complete 
page  among  the  fragments.  Several  anonymous  works  have  already 
been  mentioned,  with  some  such  title  as  'Algorismus  linealis,'  and 
possibly  this  is  one  of  them.  It  would  probably  be  possible  to  identify 
it  if  one  should  examine  the  types  and  compare  the  fragments  with 
possible  originals. 

ESTIENNE    DE    LA   ROCHE,  Villefranche. 

Ed.  pr.  1520.  Lyons,  1520. 

Born  at  Lyons,  c.  1480. 

Title.    See  Fig.  71. 

Colophon.  '  CCy  finift  larifmetique  de  maiftre  Eftienne  de  la 
roche  diet  ville  franche  natif  de  Lyon  //  fus  le  rofne.  Imprimee 
par  Maiftre  guillaume  huyon.  Pour  Conftantin  f radin  mar-//chant 
2  libraire  du  diet  Lyon.  Et  fut  acheuee  Ian  .  1 520.  le  2^.  de  Juing.' 
(F.  234,  r.). 

Description.  FoL,  17.3x25.5  cm.,  the  text  being  13x21  cm. 
I  f.  blank  +  4  ff.  unnumb.  +  230  numb.  =  235  ff.,  49  II.  Lyons, 
1520. 

Editions.  Lyons,  1520,  fol.  (here  described);  ib.,  1538,  fol. 
(see  p.  130). 

This  is  the  best  of  the  early  French  arithmetics.  Since  it  is  semi- 
mercantile  in  character,  it  was  naturally  printed  at  Lyons,  then  the  com- 
mercial center  of  France,  the  theoretical  books  being  usually  printed  at 
Paris  under  the  influence  of  the  Sorbonne.  De  la  Roche  gives  a  very 
complete  treatment  of  the  operations  with  integers,  fractions,  and  com- 
pound numbers,  and  a  large  number  of  business  applications.  Perhaps 
no  arithmetic  published  in  France  in  the  sixteenth  century  gives  a  more 
comprehensive  view  of  the  science  and  art  of  arithmetic  and  of  the 
applications  of  the  subject.  Unfortunately,  however,  de  la  Roche  took 
much  of  his  work  bodily  from  a  manuscript  of  his  master,  Chuquet, 
which  he  had  in  his  possession,  and  which  has  since  been  published. 

Other  ivorks  0/1320.  Kobel,  p.  102,  1514  ;  Peurbach,  p.  53,  1492  ; 
Raggius,  p.  98,  15  14;  Stromer,  p.  83,  1504;  Suiseth,  p.  10, tc.  1480; 
Tagliente,  p.  114,  1515  ;  Anonymous  (Tagliente  ?),  '  Libro  de  Abaco,' 


PRINTED    HOOKS 


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I30  KARA   ARITHMETICA 

Venice,  8° ;  Anonymous,  '  Libretto  de  Abaco,'  s.  1.,  8° ;  Anonymous, 
'  Algorismus  novus  de  integris,  de  minutiis  vulgaribus,  de  minutiis 
physicis,'  Augsburg.  To  an  edition  of  Sacrobosco's  Sphere  the  com- 
mentator, Johannes  Guyion,  prefixed  a  treatise  on  arithmetic,  'De 
quantitate  discreta,'  Avignon,  s.  a.,  c.  1520.  There  was  pubHshed, 
s.  1.  a.,  possibly  at  Lyons  in  this  year,  '  Le  liure  des  gectz  grande- 
ment  profitable  pour  messeigneurs  les  marchans  et  aultres,'  4°,  a  work 
on  counter  reckoning  (see  p.  7). 

ESTIENNE   DE    LA    ROCHE,  Villefranche. 

Ed.  pr.  1520.  Lyons,  1538. 

See  p.  128. 

Title.  '  Larifmetique  &  Geometrie  de  maiftre  //  Eftienne  de 
la  Roche  diet  Ville  Fran//che,  Nouuellement  Imprimee  &  des 
faultes  corrigee,//a  la  qvelle  font  adiouftees  les  Tables  de  diuers 
comptes,  auec  leurs  Ca-//nons,  calculees  par  Gilles  Huguetan 
natif  de  Lyon,  Par  lefquelles  on  pourra  facil-//lement  trouuer  les 
comptes  tons  faictz,  tant  des  achatz  que  uentes  de  toutes  mar-// 
chandifes.  Et  principalement  des  marchandifes  que  fe  uendent, 
ou  achetent  a  la  //  mefure,  come  a  Laulne,  a  la  Canne,  a  la  Toyfe, 
a  la  Palme,  au  Pied,  &  aultres  fem-//blables.  Au  poix,  come  a  la 
Liure,  au  Quintal,  au  Millier,  a  la  Charge,  au  Marc,//  &  a  Lonce, 
a  la  Piece,  au  Nobre,  a  la  Douzaine,  a  la  Groffe,  au  cent,  &  au 
Millier.//  Auec  deux  Tables  feruantz  aulx  Librayres  uendeurs 
&  acheteurs  de  papier.  En-//femble  une  Table  de  defpence,  a 
fcauoir  a  tant  pour  iour,  combien  on  defped  Lan  //  &  le  Moys, 
&  a  tant  le  moys,  combien  reuient  lan  &  le  iour,  &  a  tant  pour 
an,  co-//bien  on  defpend  tous  les  moys,  &  a  combien  reuient 
pour  chafcun  iour.//  Davantaige,  les  Tables  du  fin  dor  &  dargent, 
pour  fcauoir  (fcelon  que  le  Marc  de  billon  tiendre//daloy,  ou  de 
fin)  combien  il  uauldra  de  poix  de  fin  or,  ou  dargent  fin.//  On 
les  uend  .  .  .  lenfeigne  de  la  Sphaere,// cheulx  Gilles  &  Jacques 
Huguetan  freres.//  1538.'     (F.  i,  r.) 

ColopJion.  '  CCy  finift  Larifmetique  j  Geometrie  de  maiftre 
Eftienne  de  la  Roche  diet  Villefranche //Imprime  a  Lyon  par 
maiftre  Jacques  myt  Lan.  1538.'     (F.   160,  r.) 


PRINTED    BOOKS  131 

Description.    Fol.,  21.  i  x  33  cm.,  the  text  beinj;  16.3  x  30.6 
cm.    2  ff.  unnumb.  +  158  numb.  =  160  ff.,  59-60  11.    Lyons,  i  538. 
Editions.    See  p.  128. 
The  Huguetan  referred  to  is  the  one  mentioned  on  p.  i88, 

B^DA.     Ed.  pr.  1521.  Basel,  1563. 

'  The  Venerable  Bede  '  was  born  in  England,  probably  near  Wearmouth, 
Durham,  c.  673,  and  died  in  735.  He  was  the  most  distinguished  scholar  of 
his  time,  and  his  works  cover  all  the  branches  of  learning  then  known. 

Title.  '  Opera  //  Bedae  //  Venerabi-//lis  Presbyte-//ri,  Anglo- 
saxonis:  vi-//ri  in  divinis  atqve  hv-//manis  litcris  exercitatifsimi: 
omnia  in  octo  to-//mos  diftincta,prout  ftatim  poft  Praefa-//tionem 
fuo  Elencho  enu-//merantur.//Addit()Rerum  &  Verborum  Indice 
//copiorifsimo.//  Cum  Ca^fareae  Maieftatis  gratia  &  priuile-//gio, 
Regisque  Galliarum  ad //decennium.//Basileae,  per  loannem // 
Heruagium,  Anno  M.  D.  LXIII.'  (Surrounded  by  an  elaborate 
woodcut  with  inscriptions.)     (F.  i,  r.) 

Description.  Fol.,  24.2  X  38  cm.,  the  text  being  17.7  x  28.6 
cm.,  printed  in  double  columns,  each  8.5  cm.  wide,  61  11.  152 
pp.  unnumb.  +  271  numbered  by  columns  (i.e.,  2  numbers  to  a 
page)  =423  pp.  in  vol.  i.  8  vols,  bound  in  4.  Basel,  1563. 
Only  the  first  volume,  containingthe  arithmetic,  is  described  here. 

Editions.  Some  of  his  arithmetical  work  is  said  to  have  been 
published  in  i  5  2 1 ,  fol. ;  i  5  2  5  (in  part,  see  p.  140)  11529  (in  part, 
seep.  159);  Paris,  1544-45  (first  edition  of  the  Opera);  ib.,1554; 
Basel,  1563,  fol.  (here  described).  See  also  c.  15  15,  anonymous. 
His  '  Historia  Ecclesiastica'  appeared  as  early  as  1473. 

The  first  volume  contains  the  '  De  Arithmeticis  nvmeris  liber '  (cols. 
98-1 16),  with  little  save  an  elaborate  multiplication  table  and  a  dialogue 
on  number,  names,  and  symbols  ;  '  De  Arithmeticis  proportionibvs'  (cols. 
133-146),  with  the  '  Propofitiones  ad  acuendos  iuuenes'  often  attributed 
to  Alcuin,  but  certainly  not  Bffida's ;  *  De  ratione  calcvli '  (cols.  147- 
158),  chiefly  multiplication  tables  of  Roman  money;  '  De  nvmerorvm 
divisione  '  (cols.  159-163)  ;  '  De  loqvela  per  gestvm  digitorvm,  et  tem- 
porvm  ratione'  (cols.  1 64-1 81),  or,  as  the  headline  states  it,  'De  indigi- 
tatione,'  giving  us  almost  our  only  knowledge  of  the  finger  reckoning  or 
symbolism  of  the  Middle  Ages  in  western  Europe,  and  possibly  spurious  ; 


132  RARA   ARITHMETICA 

'  De  ratione  vnciarvm  '  (cols.  182-184),  a  treatise  on  Roman  fractions  ; 
an  extensive  treatment  of  the  calendar  and  the  computus,  with  a  descrip- 
tion of  the  astrolabe. 

FRANCESCO   GHALIGAI.     Ed.  pr.  1521.     Florence,  1552. 

A  Florentine  arithmetician  of  the  first  part  of  the  sixteenth  century.  He 
died  February  10,  1536. 

Title.  '  Practica  //  d' Arithmetica.  //  di  //  Francesco  Ghaligai 
//Fiorentino.//Nuouamente  Riuifta,  &  con  fomma // Diligenza 
Riftampata.// In  Firenze // Appreffo  i  Givnti // M.  D.  LII.' 
(F.   i,r.) 

ColopJion.  '  In  Firenze  //  Appreffo  i  Givnti  //  M.  D.  LII.' 
(F.  114,  r.) 

Description.  4°,  15.2  X  20.3  cm.,  the  text  being  12  x  17.3  cm. 
2  ff.  unnumb.  +  112  numb.  =  1 14  ff.,  37-38  11.    Florence,  1552. 

Editions.  Florence,  1521,  4°  (see  Boncompagni's  Bidletino, 
VII,  486;  XIII,  249)  ;  ib.,i548,4°;  ib.,1552,  4°  (here  described). 
The  1 521  edition  is  entitled  '  Summa  De  Arithmetica,'  but  it 
is  the  same  as  the  1552  edition  here  described.  Some  bibli- 
ographers mention  other  editions,  as  of  1540,  1551,  1562,  1572, 
1582,  I  591,  but  Boncompagni's  careful  investigation,  supported 
by  Riccardi's,  throws  doubt  upon  all  these. 

The  book  is  written  in  the  general  style  of  the  Italian  works  of  the 
sixteenth  century,  more  or  less  resembling  Borghi  (p.  16).  It  was 
intended  for  the  use  of  merchants,  and  contains  a  large  number  of 
practical  problems  showing  the  conditions  of  trade  at  the  time  of  its 
publication.  Books  10  to  13  relate  to  algebra,  and  their  chief  interest 
attaches  to  the  symbolism  employed. 

Other  works  0/1521.  Boethius,  p.  31,  1488  ;  Grammateus,  p.  124, 
15  18  ;  Paxi,  p.  77,  1503  ;  Sacrobosco,  p.  32,  1488. 

CUTHBERT   TONSTALL.     Ed.  pr.  1522.        London,  1522. 

TuNST.\LL.  Born  at  Hackforth,  Yorkshire,  in  1474;  died  November  iS, 
1559.  He  was  educated  at  Oxford,  Cambridge,  and  Padua,  was  a  man  of 
great  learning  and  energy,  and  held  important  positions  in  the  Church  and 
State.    He  was  bishop  of  London,  and  later  of  Durham. 

Title.    '  De  arte  svppvtandi  //  libri  qvattvor  //  Cvtheberti  // 

Tonstalli.'    (Surrounded  by  a  woodcut.    F.  i,  r.    See  Fig.  72.) 


PRINIKI)    UOOKS 


133 


kS^XSx 


Fig.  72.   Title  page  of  the  first  ld.iion  of  ionstall 


134  RARA   ARITHMETICA 

Colophon.  '  Impress.  Londini  in  aedibvs  Ri-//chardi  Pynsoni. 
Anno  ver-//bi  incarnati  .M.D.XXIIV/Pridie  idvs  octo-//bris. 
Cvm  pri-//vilegio  a // rege  in-//dvl-//to.'     (F.  202,  r.) 

Description.  4°,  157  X  20.9  cm.,  the  text  being  10.5  X  16.4 
cm.     202  ff.  unnumb.,  29  11.     London,  1522. 

Editions.  London,  1522,  4°  (here  described);  Paris,  1529, 
4°  (p.  135);  ib.,  1535;  ib.,  1538,  4°  (p.  135);  Strasburg,  1543; 
ib.,  1544,  8°  (p.  136);  ib.,  1548;  ib.,  1551. 

This  is  the  first  edition  of  the  first  book  wholly  on  arithmetic  that 
was  printed  in  England.  (But  see  p.  10,  1480.)  In  the  dedicatory 
epistle  Tonstall  states  that  in  his  dealing  with  certain  goldsmiths  he 
suspected  that  their  accounts  were  incorrect,  and  he  therefore  renewed 
his  study  of  arithmetic  so  as  to  check  their  figures.  On  his  appoint- 
ment to  the  See  of  London  he  bade  farewell  to  the  sciences  by  printing 
this  book  in  order  that  others  might  have  the  benefit  of  a  work  which 
he  had  prepared  for  his  own  use.  The  treatise  is  in  Latin,  and,  although 
it  was  written  for  the  purpose  of  supplying  a  practical  handbook,  is 
very  prolix  and  was  not  suited  to  the  needs  of  the  mercantile  class. 
It  is  confessedly  based  upon  Italian  models,  and  it  is  apparent  that 
Tonstall  must  have  known,  from  his  residence  in  Padua  and  his  vari- 
ous visits  to  Italy,  the  works  of  the  leading  Italian  writers.  The  book 
includes  many  business  applications  of  the  day,  such  as  partnership, 
profit  and  loss,  and  exchange.  It  also  includes  the  rule  of  false,  the  rule 
of  three,  and  numerous  applications  of  these  and  other  rules.  It  is,  how- 
ever, the  work  of  a  scholar  and  a  classicist  rather  than  a  business  man. 

The  word  '  supputandi,'  in  the  title,  was  not  uncommon  at  that  time. 
Indeed  there  was  some  tendency  to  use  the  name  '  supputation '  for 
arithmetic  and  to  speak  of  calculations  as  '  supputations.' 

Tonstall  dedicates  the  work  to  his  friend  Sir  Thomas  More,  whose 
talented  daughter  Erasmus  addressed  as  '  Margareta  Ropera  Britanniae 
tuse  decus,'— ornament  of  thine  England.  More  speaks  of  Tonstall 
in  the  opening  lines  of  his  Utopia  :  '  I  was  colleague  and  companion 
to  that  incomparable  man  Cuthbert  Tonstal,  whom  the  king  with  such 
universal  applause  lately  made  Master  of  the  Rolls  ;  but  of  whom  I  will 
say  nothing ;  not  because  I  fear  that  the  testimony  of  a  friend  will  be 
suspected,  but  rather  because  his  learning  and  virtues  are  too  great 
for  me  to  do  them  justice,  and  so  well  known,  that  they  need  not  my 
commendation  unless  I  would,  according  to  the  proverb,  "  Show  the  sun 
with  a  lanthorn.'"  The  Utopia  was  first  printed  in  15 16,  so  this  sono- 
rous praise  was  written  some  years  before  Tonstall's  arithmetic  appeared. 


PRINTED    BOOKS  135 

Some  idea  of  the  prolixity  of  the  treatise  may  be  obtained  from  the 
number  of  closely-printed  quarto  pages  assigned  to  certain  topics.  The 
chapter  '  De  Numeratione  '  fills  11  pages,  '  De  Additione  '  14  pages, 
'De  Svbdvctione '  15  pages,  '  De  mvltiplicatione  '  14  pages,  '  De,  parti- 
tione '  27  pages  (the  old  galley  method  being  used  exclusively),  and  so 
on  for  the  other  subjects.  Some  66  pages,  for  example,  are  given  to  the 
theory  of  ratio  and  proportion. 

The  title  page  was  engraved  by  Hol1)ein,  and  was  evidently  printed 
after  the  book  was  completed,  because  in  this  copy  the  errata  appear 
on  the  reverse  of  the  first  folio.  The  work  was  printed  by  Richard 
Pynson,  the  successor  to  Caxton. 

Following  the  arithmetic  is  an  appendix  :  '  Appendix  ex  Bvdaei  libro 
de  as-//fe  excerpta  :  in  qua  prifca  Latinorum  et  (irecoru  // fupputatio, 
ad  ?eftimationem  pecunie,  tum  Gallicae,//  tum  Angli-//cae  reuocatur.' 

Tonstall  also  published  a  work  in  15 18:  *  In  Lavdem  matrimonii 
oratio,'  second  edition  in  15  19,  now  very  rare. 

CUTHBERT   TONSTALL.     Ed.  pr.  1522.  Paris,  1529. 

See  p.  132. 

Title.  'De  arte  svppvtandi  libri  qvatvor//Cvthberti  Tonstalli. 
(Picture  of  a  tree  from  which  is  falling  a  broken  branch,  and  the 
words:  Noli  altum  fa//pere,  fed  time.)  Parisiis  ex  ol^cina  Roberti 
Stephani//M.D.XXIX.'     (P.  i.) 

Colophon.  '  Parisiis //excvdebat  Robertvs  Stepha-//nvs.  Ann. 
M.D.XXIX.  Prid.id.ivn.'   (P.  279.) 

Dcscriptioi.  4°,  13  X  18.8  cm.,  the  text  being  9.4  x  15.9  cm. 
271  pp.  numb. +  8  unnumb.  +  2  blank  =  281  pp.,  36-38  11.  Paris, 
1529. 

Editions.    See  p.  134. 

The  text  is  the  same  as  in  the  first  edition  of  1522.  There  has 
been  added,  however,  a  second  appendix  with  the  following  title  : 
'  Gvlielmi  Bvdaei  Parisiensis,//  secretarii  regii,  breviari-//vm  de  asse.' 

CUTHBERT   TONSTALL.     Ed.  pr.  1522.  Paris,  1538. 

See  p.  132. 
Title.    '  De  arte  svp-//putandi  libri  qua-//tuor,  Cutheberti  // 
Tonftalli.//  (Large  woodcut.)   Parisiis.//  Ex  officina  Roberti  Ste- 
phani.//M.D.XXXVin.'     (P.  i.) 


136  RARA   ARITHMETICA 

Colophon.  '  Excvdebat  Robertvs  Stephanvs  Parisiis,  //  aim. 
M.  D.  XXXVIII. //xvi.  cal.  novemb.'     (P.  259.) 

Description.  4°,  14.4  x  20.1  cm.,  the  text  being  9.5  x  16  cm. 
259  pp.  mmib.,  39  11.     Paris,  1538. 

See  p.  134. 

CUTHBERT   TONSTALL.     Ed.  pr.  1522.    Strasburg,  1544. 

See  p.  132. 

Title.  'De  arte  //svppvtan//di,  libri  qvatvor//Cvthberti  Ton- 
stalli,//hactenus  in  Germania  nus-//quam  ita  impreffi.// loan. 
Stvrmivs.// Arithmeticam  Cvthbertvs // Tonftallus  prae  cseteris 
dilucide  &  pure  tradidit:  atq5//ita  tradidit,  ut  ars  ipfa  dum  hie 
author  extat,  con-//tenta  fcriptore,  doctorem  non  maximopere 
ahquem  //  requirat.  Non  nego,  poffe  ex  alijs  quoque  difci :  //  fed 
hie  docet  erudite,  perfpicue  latine,  id  quod  non  fa//ciunt  caeteri: 
nee  abest  longe  a  perfectione,  qui  eius  //  praecepta  intelligit.// 
Argentorati,  ex  offi.// Knobloch.  per  Georg.  Machaerop.'    (P.  i.) 

Colophon.  '  Argentorati,  ex  officina//Knoblochiana,  per  Gq-/ / 
orgivm  Machaero-//poevm,  mense  //  febrvario  //  anno,//  M.  D. 
XLIIIL'     (P.  478.) 

Description.  8°,  9.5  X  15  cm.,  the  text  being  6.8  X  11.4  cm. 
25  pp.  blank +  453  numb.  =  478  pp.,  26  11.  Strasburg,  1544. 
This  copy  is  bound  with  the  arithmetic  of  Victorius  Strigelius. 

Editions.  See  p.  134.  This  is  the  same  as  the  edition  of  1529, 
having  the  second  appendix  there  mentioned.  It  is  the  second 
Strasburg  edition.  It  is  interesting  to  see  that  the  classical 
influence  on  the  Continent  was  such  that  seven  out  of  the  eight 
editions  appeared  in  Paris  or  Strasburg. 

JOHANNES  AVENTINUS. 

Ed.  pr.  1522.  Regensburg,  1532. 

Thurnmayer.    Born  at  Abensberg,  Bavaria,  July  4,  1477  ;  died  at  Regens- 
burg, January  9,  1534.     He  wrote  on  history. 

Title.    See  Fig.  73. 

Colophon.  '  Ratifpon^  apud  loannem  Khol  //  Anno.  MD- 
XXXII.'  (Large  woodcut  and  date,  'lo.  Kol  1532.')   (F.  12,  r.) 


PRINTED    HOOKS  137 

Description.  4°,  14.2  x  19.5  cm.,  the  Icxt  bcin^^  9.9  x  14  cm. 
12  ff.  unnumb.,  26  11.     Rcgensburg,  1532. 

Editions.  Niirnberg,  1522;  Regensburg,  1532,  4°  (here 
described). 

The  book  is  primarily  a  treatise  on  numerical  finger  symbolism,  and 
contains  the  most  complete  explanation  of  that  subject  extant.    It  gives 

'^ABACVSAT'-?*' 

QVE  VETVSTISSIMA,  VETERVM 

larinorum  per  digitos  manufcp  numc^ 

randi(quinetiam  loqucndi)  cofuc^ 

tudo^Exbcdacupiduris  &  ima^ 

ginibus,  inucta  rcgmoburgrj 

fiue  rsctobonaPjin  biblio^ 

thcca  diui  hacmtrani, 

Atcp  hoc  conuc^ 

tu  auguftali 

Rcucrcndi 

Atcp  docfliflimi  Domini  Lucac  bonfrj  dc<^ 

cani  patauini  fccrctarrj  Rcucrcndiili^ 

miCardinalis  Laurcnti]  Campcgf] 

zc»  Aufpicijs  A  Io»  Aucn/ 

tino  Edica* 


Gcrmania  lllufiranda. 

Fig.  73.    Title  page  of  the   1532  .aventints 

illustrations  showing  the  representation  of  the  numbers  up  to  one  million 
by  means  of  the  fingers  and  arms  (see  Fig.  74).  This  finger  symbolism 
is  found  in  the  works  of  Baeda,  it  was  practical  in  both  the  East  and 


138 


RARA   ARITHMETICA 


West  during  the  Middle  Ages,  and  it  is  mentioned  by  several  sixteenth- 
century  arithmeticians.    (Compare  Fig.  74  with  Fig.  35,  p.  57.) 


Fig.  74.    Finger  symbolism  from  aventinus 


ADAM    RIESE.     Ed.  pr.  1522. 


Leipzig,  1538. 


Ryse,  Ris,  Ries.  Born  at  Staffelstein,  near  Bamberg,  c.  1489;  died  at 
Annaberg,  March  30,  1559.  One  of  the  most  celebrated  Rechenmeistero 
of  the  sixteenth  century,  and  the  most  influential  of  all  the  Germans  in 
replacing  the  coimter  reckoning  ('  auff  der  Linien  ')  by  the  written  computa- 
tions ('  auff  Federn '). 

Title.  '  Rechnung  auff  //der  Linien  vnd  Federn  //Auff  aller- 
ley  handtirung  ge-//macht/ durch  Adam  Rifen.// (Woodcut  of 
counting  house,  with  reckoning  on  the  line  abacus.)  Item  auffs 
new  vberfehen  vnd  //an  viel  orten  gebeffert.//  M.DXXXVIIL' 
(E  i,r.) 

ColopJion.  '  Gedruckt  zu  Leiptzigk  //  durch  Valentinum  // 
Schumann.     1538.'     (F.  63,  v.) 

Description.  8°,  9.7  X  15.1  cm.,  the  text  being  7.2  x  1 1.5  cm. 
63  ff.  unnumb.  +  2  blank  (with  manuscript  notes)  =65  ff.,  28- 
33  11.     Leipzig,  1538. 

Editions.  In  order  to  make  clear  the  different  editions  of 
Riese's  work,  it  is  necessary  to  distinguish  between  the  four 
arithmetics  which  he  published.    These  were  as  follows  : 


PRINTED    BOOKS  139 

1.  '  Rcchnung  aiiff  der  linihen  gemacht  durch  Adam  Riesen 
vonn  Staffelsteyn // in  massen  man  cs  pflegt  tzu  Icrn  in  alien 
rechenschulen  gruntlich  begriffen  anno  1 5  1 8.'  (Graessc,  followed 
by  Unger,  p.  50,  who  knew  of  no  extant  copy.)  A  second  edi- 
tion appeared  in  1525  (' Getruckt  tzu  Erffordt  durch  Mathes 
Maler  M.  CCCCCxxv  Jar,'  8°,  43  ff.),  and  a  third  in  1527.  This 
was  embodied  in  his  second  arithmetic  which  is  here  described. 
The  work  is  rare. 

2.  '  Rechnung  auff//der  Linien  vnd  Federn,'  Riese's  best- 
known  work.  The  title  of  the  first  (i 522)  edition  was  as  follows: 
'  Rechenung  auff  der  linihen  vnd  federn  in  zal/mafs/vnd  gewicht 
auff  allerley  handierung/  gemacht  vnd  zusamen  gelesen  durch 
Adam  Riesen  von  Staffelstein  Rechenmeister  zu  Erffurdt  im 
1522  Jar.  Itzt  vff  sant  Annabergk  durch  in  fleyssig  vbersehen/ 
vnd  alle  gebrechen  eygentlich  gerechtfertigt/ vnd  zum  letzten 
eine  hiibsche  vnderrichtung  angehengt.'  The  following  editions 
of  this  work  are  known  to  me:  Erfurt,  1522;  ib.,  1525,  8°; 
Niirnberg,  1527;  1528,  8°;  Erfurt,  1529,  8°;  ib.,  1530;  Leip- 
zig, 1533,  8°  (first  edition  containing  Helm's  Visirbuch ;  see  p. 
142);  Frankfort,  1535,  12°;  Annaberg,  1535,  8°  (p.  141);  1536, 
12°;  Leipzig,  1538,  8°  (here  described);  1541;  Frankfort,  1544, 
8°;  Leipzig,  1544;  s.  1.,  1548,  8°;  Leipzig,  1548,8°;  ib.,  1550; 
Breslau,  1550;  Frankfort,  1552,  8°;  Leipzig,  1554;  ISS^,  12°; 
Frankfort,  1558,  8°  (p.  141);  Leipzig,  1562,  12°;  Frankfort, 
1564,  8°;  Frankfort,  1565,  8°  (p.  142);  Frankfort  (a.  Oder  .?), 
1568,  8°;  Stettin  (Frankfort.?),  1570,  8°;  Leipzig,  1571,  8° 
(p.  142);  Frankfort,  1574;  Magdeburg,  1579,  8°;  Frankfort, 
1 58 1,  12°;  ib.,  1585,  8°;  Leipzig,  1586;  Frankfort,  1586; 
Wittenberg,  1587;  Niirnberg,  1592,  8°;  Frankfort,  1592,  8° 
(p.  143);  Leipzig,  1598,  12°.  There  were  several  editions  after 
1600.  It  is  possible  that  some  of  the  editions  here  mentioned 
may  be  of  Riese's  fourth  book,  the  titles  being  much  alike  and 
printers  varying  them  from  time  to  time. 

3.  *  Ein  Gerechent  Biichlein,'  first  published  at  Leipzig  in 
1533;   second  edition  in  1536.     See  p.  171. 


I40  KARA   ARITHMETICA 

4.  '  Rechnung  nach  der  lenge/auff  den  Linihen  vnd  Feder,' 
first  published  in  1550,  4°.    See  p.  250. 

Kuckuck's  remark,  that  over  twenty-six  editions  of  Riese's 
arithmetics  appeared,  greatly  underestimates  the  number.  More 
than  forty  appeared  in  the  sixteenth  century  alone,  and  several 
were  published  in  the  seventeenth  century. 

This  was  probably  the  most  popular  commercial  arithmetic  of  the 
sixteenth  century.  So  firmly  did  it  impress  itself  upon  the  schools  that 
'  nach  Adam  Riese  '  is  a  common  expression  in  Germany  to-day,  nearly 
four  hundred  years  after  the  first  of  his  books  appeared.  It  was  to 
Germany  what  Borghi's  book  was  to  Italy  and  Recorde's  to  England. 
It  differed  from  Riese's  first  book  in  that  it  emphasized  computation 
by  the  aid  of  the  Hindu-Arabic  numerals  instead  of  the  counters. 
There  is  no  other  book  that  gives  as  good  a  picture  of  the  sixteenth- 
century  mercantile  problems  of  Germany,  and  of  the  methods  of 
solving  them. 

Other  works  0/1522.  Albert  of  Saxony,  p.  9,  c.  1478  ;  Boethius, 
p.  27,  1488;  Budaeus,  p.  99,  15 14;  Kobel,  p.  102,  15  14;  Ortega,  p. 
93,  15  12  ;  Sacrobosco,  p.  32,  1488;  Francisco  Pelacani,  '  Arithmetica 
prattica,'  Florence;  Ludovico  Vincento  (Vincentino)  degl'  Arrighi,  'La 
operina  .  .  .  da  .  .  .  bellissima  Ragione  di  Abbacho,'  Rome,  4°,  with 
editions  at  Venice  in  1532,  1533,  chiefly  on  chirography. 

Works  0/1323.  Borriglione,  p.  86,  1506;  Ciruelo,  p.  60,  1495; 
Grammateus,  p.  124,  1518;  Paciuolo,  p.  54,  1494;  Reisch,  p.  82, 
1503;  Sacrobosco,  p.  35,  1488;  Tagliente,  p.  114,  15  15.  Rodrigo  Fer- 
nandez de  Santaella  (or  Valencia,  see  p.  269,  1555),  '  Ars  coputandi,' 
Saragossa,  fol.  There  was  also  written  in  this  year,  but  published  s.  1.  a., 
a  work  by  Vincenzo  Barziza  entitled  '  Operetta  nouamente  composta,' 
8°,  39  ff.,  containing  a  few  mercantile  rules  and  tables. 

Works  0/1524.    Feliciano,  p.  145,  1526  ;  Tagliente,  p.  115,  15 15. 

Works  0/1525.  Herodianus,  p.  60,  1495  ;  Kobel,  p.  102,  15 14; 
Riese,  p.  139,  1522  ;  Rudolff,  p.  151,  1526;  Tagliente,  p.  114,  15 15  ; 
Bede  et  al.,  '  Valetius  Probus  et  Petrus  Diaconus  de  notis  Romanorum, 
Demettius  Alabaldus  de  minutiis,  ponderibus  et  mensuris,  Ven.  Beda  de 
computo  per  gestum  digitorum,'  etc.,  Venice,  4°;  Angelus  Mutinens 
(i.e.  of  Modena),  'Thesavro  de  Scrittori  opera  artificiosa  le  quale  con 
grandissima  arte,  si  per  pratica  come  per  geometria  insegna  .  ,  .,'  s.  1., 
M.D.XXxv,  has  four  folios  at  the  end  relating  to  arithmetic,  with  the 
note  '  Angelus  Mutinens  composuit,'  and  there  seems  to  have  been 
an  edition  in    1525,  and  another  s.  a.  pubHshed  at  Rome;  Giovanni 


PRINTl-:!)    HOOKS  141 

iagliente  published  two  editions  of  a  work  <  hicfly  on  Ijookkecping,  4°, 
24  ff.,  Venice,  beginning,  '  Considerando  io  loanni  Taiente  ciuanlo  e 
necessaria  cosa  ali  nostri  raagnifici  gelilhomeni  &  ad  altri  mercatanti.' 

ADAM    RIESE.     Ed.  pr.   1522.  Annabcrg,  1535. 

See  p.  1  j;S. 

Title.    The  title  pai;e  is  missing. 

Coloplioji.  'CNach  difer  vnderrichtung  kanftu  auffs  be//hen- 
deft  alle  Exempel  in  der  Ealii  machen :  W61//left  folch  Biich- 
lin  vnnd  kurtze  erklerung  ietzt/  //welches  ich  zum  andern  mal 
laffe  aufzge-//hen/  zu  danck  an  nemen/  wil  ich  ver-//dienen/ 
vnd  dir  auffs  eheft  ich  //  mag  die  Practica  nach  al-//lcm  fleifz 
heraufzftrei//chen.  Datum //auff  fanct  //  Annaberg/  //  Dinftag 
nach // Martini.  Im  Iar//M.  D.  XXV.'  (F.  55,  r.)  On  f.  69, 
v.,  is  the  following  colophon:  '  Alfo  ift  kiirtzlich // befchriben 
vnd  //  begriffen  //  die  Confection  //  der  Vifier  riitcn  mit  //  Irer 
ubung  vnd  gebrauch.//  CEnd://  An.  M.  D.  XXXV.//  Im  Christ- 
monat.'  This  latter  is  evidently  the  date  of  printing  of  the  entire 
book. 

Description.  8°,  8.8  x  13.2  cm.,  the  text  being  6.8  x  1 1.6  cm. 
69  ff.  unnumb.,  31  11.     Annaberg,  1535. 

Editions.    See  p.  139. 

This  contains  the  '  Vifirbflchlin '  of  Erhart  Helm,  as  in  the  1533 
edition,  but  it  does  not  give  his  name.    See  the  1565  edition  (p.  142). 

ADAM    RIESE.     Ed.  pr.  1522.  Frankfort,  1558. 

See  p.  13S. 

Title.  The  title  page  is  missing.  Page  numbered  2  begins  as 
follows  :   '  Vorrede  in  diO  Rechen-//buch/ Adam  Rifen.' 

Colophon.  'CEnd.//Zu  Franckfurt  bei  Chr.  Egeb.  erben/// 
Anno  1558.'     (F.  numb.  87,  r.) 

Description.  8°,  9.4  X  15  cm.,  the  text  being  6.5  x  12.2  cm. 
87  ff.  numb.  +  I  blank  =  88  ff.,  31  11.     Frankfort,  1558. 

Editions.    See  p.  139. 

This  contains  the  '  Vifirbiichlin '  of  Erhart  Helm,  as  in  the  1533 
ei  lit  ion. 


142  RARA   ARITHMETICA 

ADAM    RIESE.     Ed.  pr.  1522.  Frankfort,  1565. 

See  p.  1 38. 

Title.  '  Rechenbuch/  Vff  Lini//en  vnnd  Ziphren/  In  allerley 
// Handtierung/ Gefchefften  vnd  Kauff-//mannfchafft.  Durch 
Adam  //  Rifen.//  Mit  new  en  kunftlichen  Regeln  vnd  Ex-// 
emplen  gemehrt/  Innhalt  furge-//ftelten  RegiftersV/Vifier  vnd 
Wechfelruten  kunftlich  vnd  //  gerecht  zumachen/  aul?  dem  Qua- 
drat/ //  Durch  die  Arithmetic  vnd  Geometri.  //  Von  Erhart 
Helm/  Mathema//tico  zu  Franckfurt/  be  //  fchriben.// Alles  von 
newem  jetzund  widerumb  erfe-//hen  vnd  corrigirt.// (Woodcut  of 
counting  house)  Franck,  Bei  Chr.  Egen.  Erben.  1565.'    {F.i,r.) 

Colophon.    'M.  D.LXV;    (F.  113,  r.) 

Description.  8°,  9.3  X  i  5.5  cm.,  the  text  being  6.5  X  1 1.9  cm. 
105  ff.  numb.  +  8  unnumb.  =  113  ff.,  28  11.     Frankfort,  1565. 

Editions.    See  p.  139. 

See  p.  140.  The  Visirbuch  of  Helm  begins,  with  no  separate  title 
page,  on  f.  7 7,  V. :  '  Vifirbi^chlin  //  Hernach  folget  der  ware  //  Procefz/ 
vnnd  kurtzeft  weg/  wie  mann  Vifir  riilhen  machen  fol/  aufz  dem  // 
Quadraten/ Auff  alle  Eich.'  The  name  of  the  author,  'Erhart  Helmen,' 
appears  in  the  headlines  of  each  folio  recto.  This  part  of  the  book  is 
strictly  speaking  not  an  arithmetic,  but  it  includes  a  few  explanations  of 
those  processes  that  are  necessary  in  gauging.  It  also  includes  a  table 
of  square  roots  to  the  equivalent  of  three  decimal  places,  and  a  brief 
explanation  of  roots.    It  was  published  in  separate  form  in  1529. 

ADAM    RIESE.     Ed.  pr.  1522.  Leipzig,  1571. 

See  p.  138. 

Title.  '  Rechnung  auff  //  der  Linien  vnd  Federn/  //  auff 
allerley  Handtierung/  //Gemacht  durch //Adam  Rifen.// (Wood- 
cut of  Adam  Riese,  with  motto  :  '  Anno  1550  Adam  Ries  Seins 
Alters  Im  LVIII.')  Auffs  newe  durchelefen/ vnd //zu  recht 
bracht.//  1571.'     (F.  i,  r.) 

Colophon.  'Zu  Leipzig  druckts  //  Hans  Rhambaw/  //Im  Jar// 
1 571.'     (F.  94,  V.) 

Description.  8°,  9.7  x  i  5.6  cm.,  the  text  being  6.9  X  1 1.8  cm. 
94  ff.  unnumb.,  24  11.     Leipzig,  1571. 


PRINTED    BOOKS  143 

Editions.    Sec  p.   i  39. 

See  p.  140.  This  is  substantially  identical  with  the  iS.^S  edition 
except  as  to  the  title  page. 

ADAM    RIESE.     Ed.  pr.  1522.  Frankfort,  1592 

See  p.  138. 

Title.    See  Fig.  75. 

Description.    8°,  9.5  X  15.5  cm.,  the  text  being  6.7  x  1 1.8  cm. 

1  f.  unnumb.  +  79  numb.  =80  ff.,  26-27  11.     Frankfort,   1592. 

Editions.  See  p.  1 39.  Bound  with  this  is  Helm's  work  of  1 592 
(described  later). 

FRANCESCO    DAL   SOLE.     Ed.  pr.  1526.      Ferrara,  1546. 

A  French  arithmetician,  born  c.  1490,  and  living  in  Kerrara  at  the  time  of 
writing  his  books. 

Title.    See  Fig.  ^6. 

Colophon.  '  In  Ferrara  Nella  Stampa  di  .M.  Giouanni  de  bugl- 
hat  &.  M.  Antonio  //  Hucher  Compagni,  Ad  Inftantia  de  .M. 
Rinaldo,  cuoco  dello  Illuftriffi-//mo  fignor  Duca,  nel  mefe  di 
zenaro  1546.'     (F.  42,  v.) 

Description.    4°,  14  X  19  cm.,  the  text  being  12.  i  x  16.5  cm. 

2  ff.  unnumb.  +40  numb.  =42  ff.,  31  11.     F"errara,  1546. 

Editions.  Sole  published  a  '  Libretto  di  Abaco  '  in  Venice  in 
1526,  8°,  and  this  is  merely  a  revision  of  that  work.  A  third 
edition  appeared  in  1564  (see  p.  146). 

This  is  little  more  than  a  primer  of  arithmetic.  It  contains  the  funda- 
mental operations,  a  few  of  the  more  important  applications,  eight  pages 
of  products  and  roots,  and  several  pages  on  astrology.  The  part  on  astrol- 
ogy includes  some  theory  of  the  calendar,  as  may  be  seen  by  the  title  : 
'  Incominciano  le  regoline  daftrologia,  p  ritrouare  ha  quati  di  et  minute 
fa  la  luna,  la  Ira  diiicale,  et  infinite  getileffe,  Delli  circuli,  elementi,  et 
natura,  del  monde.'  The  most  distinctive  feature  of  the  arithmetic  is 
the  combination  of  number  and  space  concepts.  For  example,  in  addi- 
tion the  author  considers  not  only  abstract  but  compound  numbers  as 
well  as  geometric  magnitudes.  ('  Regola  dellne  additioni  in  generalita, 
tanto  geometrica,  quanto  arithmetica.  Ca.  6.')  The  same  idea  is  carried 
out  in  the  other  fundamental  operations. 


144 


RARA   ARITHMETICA 


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PRINTED    BOOKS 


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146  RARA   ARITHMETICA 

FRANCESCO   DAL   SOLE.     Ed.  pr.  1526.     Ferrara,  1564. 

See  p.  143. 

Title.  '  Instrvtioni  //  et  Regvle  //  di  Francesco  //  dal  Sole,// 
Francese.//Cittadino  di  Ferrara,  Sopra  il  fon-//damento  delle 
alme  fcientie  d'Abbac-//co,  Arithmetica,  Geometria,  Cof-//nio- 
grafia,  &  Mathematica,  No-//uamente  riftampate,  &  con//par- 
ticolare  addittioni  di  //  effo  Authore,//  aggionte.//  In  Ferrara, 
Aprefso  Francefco  di  Rossi //da  Valenza.//M.  D.  LXIIL'  (F.  i ,  r.) 

Description.  4°,  14.8  X  20  cm.,  the  text  being  i  i.i  x  15.5  cm. 
4  pp.  unnumb.  +  71  numb.  =  75  pp.,  38  11.    Ferrara,  1564. 

Editions.    See  p.  143. 

Although  the  title  of  this  edition  is  quite  different  from  that  of  the 
first,  as  shown  in  Fig.  76,  the  work  is  essentially  the  same.  A  set  of  verses 
entitled  '  II  Sole '  (The  Sun),  a  play  upon  the  author's  name,  which 
appeared  in  the  first  edition,  gives  place  to  some  Latin  lines  in  this  one. 

FRANCESCO    FELICIANO  da  Lazesio. 

Ed.  pr.  1526.  Venice,  1526. 

Born  at  Lazisa,  near  Verona ;  he  was  living  in  1 563. 

Title.    See  Fig.  'j'j . 

Colophon.  '  CStampato  nella  inclita  Citta  di  Vinegia,  apreffo 
// fanto  Moyfe  nelle  cafe  nuoiie  luftiniane:  Per // Fracefco  di 
Aleffandro  Bindoni,  &  Ma-//pheo  Pafmi,  compagni.  Nelli  anni 
//  del  fignore,  1527.  Del  mefe  //  di  Zenaro.  Regnante  il  //  Sere- 
niffimo  Princi-//pe  meffer  An-//drea  Gritti.// A  B  C  D  E  F  G 
HIKLMNOPQRST  V.//Tutti  fono  duerni.'    (F.  80,  r.) 

Description.  4°,  15.1  X  20.8  cm.,  the  text  being  13.4  X  17.  i 
cm.    80  ff.  unnumb.,  41  11.    Venice,  1526. 

Editions.  Feliciano  published  two  works,  of  which  the  first 
was  entitled  '  Libro  de  Abaco,'  and  appeared  in  the  following 
editions:  Venice,  1 5  1 7  (the  colophon  date  is  15  18),  8°;  ib.,  1519, 
8°;  ib.,  1524,  8°;  ib.,  1532,  8°.  His  second  work  was  a  revision  of 
his  first,  and  is  the  one  here  described,  and  this  appeared  in  the 
followingeditions:  Venice,  i  526,4°  (here  described);  ib.,  1527,  4°; 
ib.,  1536,  4°  (p.  148);  ib.,  1545,  4°  (p.  149);  ib.,  1550,  4°  (p.  149); 


]lib:o&iaiitbmcnca'r  ^comctria 

fpccuUrtua  1  pjaticale :  Coinporto  per  macftro 
¥M        frmceko  fdicimo  oa  %^!,ifio  Ucronck 
•^titulato  Ocala  grainalddli ; 
B^ouamcnte  ITampato* 


w>nmuJi^>Ml'^»'n»>i'»>'f»f'>i'^.'StP>iJl 


JT"-'"-'-''' 


Chi  vol  aprif  vna  fcrragli  a  forte 
Jncima  dVna  Torre  ouer  Caftello 
Blfogna  haucr  la  Scala  accio  di  quello 
Giongcr  fi  pofla  alleferrate  porte 

Aprcilo  fa  meftier  Phuom  feco  portc 
CNon  hauendo  la  Chiaue  del  fugello} 
Vnferro  chcapellatoe  grimaldello 
Sinon  d'aprir  I'imprefe  fue  ficn  carte, 

Tal  quefto  mio  libretto  moftra  come 
SagUr  ^  pofla  allalte  (erraturc 
Et  quelle  aprirfcnza  faticagraue, 

Infinite  ragion,ponti,&  mifure 
Quefto  chiarilce  con  modo  fuaue 
Cui  fcala  e  grimaldello  e  il  proprio  nome. 
Ma  non  lodar  mie  come 

Co'l  tuolieto  biafmarjchel  biafmo  nafce 
Sol  dc  cui  nel  mal  dir  s^alegra  e  pafcc* 

M.  D,  XXVh 


Con  gratia 'rpmiilcgio. 


[^ 


Fig.  77.    TiiLK  i-age  ok  ikliciam 


148  RARA  ARITHMETICA 

ib.,  1560,  4°  (p.  149);  ib.,  1560,  4°  (another  edition);  ib.,  1561 
(the  colophon  of  one  1560  edition);  Verona,  1563,  8°  (p.  150); 
s.  1.  (Venice  ?),  1563,  4°;  Venice,  1570.  There  were  also  seven- 
teenth-century editions  extending  as  late  as  1692.  For  the  1602, 
1629,  and  1669  editions  see  pp.  150,  151.  This  first  edition  was 
evidently  begun  in  1526,  but  completed  in  January  1527,  as 
the  colophon  shows.    It  is  often  assigned  to  the  latter  year. 

Feliciano's  second  work  was  highly  esteemed  as  a  textbook  for 
schools.  It  follows  the  lines  laid  down  by  Borghi  (p.  16),  and  the 
author  acknowledges  his  indebtedness  to  him  and  to  Paciuolo  (p.  54). 
The  first  part  of  the  book  is  commercial  in  character,  and  in  the  second 
part  the  author  treats  of  roots,  rule  of  false,  and  algebra,  the  third  part 
iDcing  devoted  to  geometry  from  the  practical  side.  More  complete 
than  the  Treviso  book,  more  modern  than  Borghi,  more  condensed  and 
practical  than  Paciuolo,  few  books  had  greater  influence  on  the  subse- 
quent teaching  of  elementary  mathematics.  The  fanciful  name,  '  Scala 
grimaldelli,'  is  explained  in  the  verses  on  the  title  page.  Just  as  it  is 
necessary  in  attacking  a  castle  to  have  a  ladder  (scala)  and  a  skeleton 
key  (grimaldello)  to  open  locks,  so  in  attacking  mathematics  it  is  nec- 
essary to  have  a  book  that  answers  the  same  purposes. 

FRANCESCO   FELICIANO  da  Lazesio. 

Ed.  pr.  1526.  Venice,  1536. 

See  p.  145. 

Title.  '  Libro  di  Arithmetica  2  Geometria  //  fpeculatiua  2  pra- 
ticale  :  Compofto  per  maeftro  //  Francefco  feliciano  da  Lazifio 
Veronefe//Intitulato  Scala  Grimaldelli  ://Nouamente  ftampato.' 
(F.  r,  I.    The  rest  is  substantially  as  in  the  first  edition,  Fig.  j-] .) 

Colophon.  '  CStampato  nella  inclita  Citta  di  Vinegia,  apreffo 
//  fanto  Moyfe  nelle  cafe  nuoue  luftiniane  :  Per  //  Fracefco  di 
Aleffandro  Bindoni,  &  Ma-//pheo  Pafmi,  compagni.  Nelli  anni  // 
del  fignore.  1 536.  Del  mefe  //di  Zenaro.  Regnante  il//Serenifri- 
mo  Princi-//pe  meffer  An-//drea  Grittti.// ABCDEFGHIKLM 
NOPQRSTV.//Tutti  fono  duerni.'     (F.  80,  r.) 

Description.  4°,  15.6  X  20.8  cm.,  the  text  being  13.4  X  17.  i 
cm.    80  ff.  unnumb.,  41  11.    Venice,  1536. 

See  above. 


PRINTED    BOOKS  149 

FRANCESCO    FELICIANO  da  Lazesio. 

Ed.  pr.  1526.  Venice,  1545. 

See  p.  146. 
Title.    This  is  substantially  the  same  as  in  the   1526  edition 

(P-  147)- 

ColopJton.  With  the  exception  of  the  date  (1545),  this  is  sub- 
stantially as  in  the  1536  edition. 

Description.  Substantially  as  in  the  1 536  edition.  Venice,  1 545. 

Editions.    See  p.  146. 

FRANCESCO    FELICIANO  da   Lazesio. 

Ed.  pr.  1526.  Venice,  1550. 

See  p.  146. 

Title.    This  is  substantially  the  same  as  in  the  1526  edition. 

Colophon.  '  Stampato  nella  inclita  Citta  di  Vinegia,  per  // 
Francefco  Bindoni,  &  Mapheo  Pafmi,// Nelli  anni  del  noftro 
Signore.// M.  D.  L.// Registro.// A  BCDEFGHIKLM 
N  O  //  P  O  R  S  T  V.//Tutti  fonno  duerni.'     (F.  80,  r.) 

Description.  4°,  15.6  X  20.8  cm.,  the  text  being  13.4  x  17.2 
cm.    80  ff.  unnumb.,  41  11.    Venice,  15  50. 

Editions.    See  p.  146. 

FRANCESCO    FELICIANO  da  Lazesio. 

Ed.  pr.  1526.  Venice,  1560-61. 

See  p.  146. 

Title.  This  is  substantially  the  same  as  in  the  1526  edition 
(p.  147),  but  bears  the  date  M.D    LX. 

Colophon.  'Stampato  nella  Inclita  Citta  di  Vinegia,  Per  Fran- 
//cefco  de  Leno.  Nell' anno  del  N.  Signore.// M.  D.  LXL' 
(F.  79,  r.) 

Description.  4°,  15.9  X  21. i  cm.,  the  text  being  14.3  x  17.4 
cm.    79  ff.  unnumb.,  41  11.    Venice,  1560  (colophon  1561). 

Editions.  See  p.  146.  Riccardi  mentions  two  identical  editions 
of  this  year,  one  of  them,  here  described,  with  the  colophon  date 
1561. 


I50  RARA   ARITHMETICA 

FRANCISCO    FELICIANO  da  Lazesio. 

Ed.  pr.  1526.  Verona,  1563. 

See  p.  146. 

Title.    This  is  substantially  the  same  as  in  the   1561  edition. 
Description.    4°,   14.9  X  19.4  cm.,  the  text  being  13.6  X  17.1 
cm.,  41  11.,  64  ff.  imnumb.  (part  III  missing).    Verona,  1563. 
Editions.    See  p.  146. 
The  third  part,  on  geometry,  is  missing  in  this  copy. 

FRANCESCO   FELICIANO  da   Lazesio. 

Ed.  pr.  1526.  Verona,  1602. 

See  p.  146. 

Title.  This  is  substantially  the  same  as  in  the  1526  edition 
except  '  De  nuouo  riftampato,  &  da  molti  errori  corretto,  &  ac- 
crefciuto  di  molte  cofe  da  M.//  Fillipo  Marcario  Veronefe  Rafo- 
nato  publico  della  Magnifica  Citta.//Con  le  gionta  della  Regola 
del  Catain  del  medefimo.// In  Verona,  Apreffo  Dionigi  Filiberi. 
CI3  13  C  II.'    (P.  I.) 

ColopJion.  'In  Verona,// Nella  Stamparia  di  Angelo  Tamo. 
1602.'     (P.  284.) 

Description.  4°,  14.7  X  19.2  cm.,  the  text  being  11.5  X  15.7 
cm.  5  pp.  blank  +  7  unnumb.  +  276  numb.  =  288  pp.,  32  11. 
Verona,  1602. 

FRANCESCO    FELICIANO  da  Lazesio. 

Ed.  pr.  1526.  Padua,  1629. 

See  p.  146. 

Title.  '  Scalla  //  Grimaldelli  //  Libro  di  //  Aritmetica,  et  Geo- 
metria // Speculatiua,  &  Pratticale // Di  M.  Francefco  Feliciano 
Veronefe. //Diviso  in  tre  libri.//  ...  In  Padoua,  Per  Donato  Paf- 
quardi,  &  compagni.    i629.//Conlicenza  de'  Superiori.'   (F.  i,  r.) 

Description.  4°,  15  X  20.5  cm.,  the  text  being  11.4  X  16  cm. 
4  pp.  unnumb.  +  276  numb.  =  280  pp.,  32-34  11.     Padua,  1629. 

Editions.    See  p.  146. 


PRINTKI)    BOOKS  151 

FRANCESCO    FELICIANO  da  Lazesio. 

Ed.  pr.   1526.  Venice,  1669. 

See  p.  146. 

Title.  '  Scala //Grimaldelli  // libro  di  // aritmctica,  e  <;e()- 
metria // Spcculatiua,  e  Pratticale // Di  M.  l^Vancesco  Feliciano 
//Veronese.// Diviso  inTre  Libri.  ...  Di  nuouo  riftampato,  e  da 
molti  errori  corretto,  &  accrcfciuto  di  molte  cofe  da  //  M.  Filippo 
Macario  Veronefe  Rafonato  publico  della  Magnifica  //  Citta.  Con 
I'aggionta  della  Regola  del  Catain  del  mcdefimo.//  Al  Molt'  lUuf- 
tre  Signor,  e  Padron  Colendifs.  il  Signor  Gio:  Battista  Sorer.// 
Venetia,  MDCLXIX.// Preffo  Gio:  Giacomo  Hertz.'    (F.  i,  r.) 

Description.  4°,  16.  i  X  22  cm.,  the  text  being  i  1.8  X  16  cm. 
6  pp.  blank  +  6  unnumb.  +  240  =  252  pj).,  41  11.     Venice,  1669. 

Editions.    See  p.   146.  '^' 

See  p.  148.  It  speaks  well  for  this  work  of  Feliciano's  that  this  edition 
should  have  appeared  one  hundred  and  forty-three  years  after  the  book 
was   first  published. 

CHRISTOFF  RUDOLFF.     Ed.  pr.  1526.      Niirnberg,  1534. 

Born  at  Jauer  c.  1500,  but  the  dates  of  his  birth  and  death  are  unknown. 

Title.  '  Kunftliche  rech//nung  mit  der  ziffer  vnnd  mit  //  den 
zal  pfenninge // fampt-// der  Wellifchen  Practica // vnd  allerley 
vorteil // auff  die  Regel  de  Tri.// Item  vergleichug  mancher-// 
ley  Land  vii  Stet  //  gewicht/  Elnmas//Muntz  ec.  AUes  durch  // 
Chriftoffen  Rudolff  zu/ Wein  verfertiger.//  1534.'     (F.  i,  r.) 

Colophon.  '  Getrukt  zu  Nurmberg  bey//Johan  Petreio//im 
iar  nach  //  der  geburt  Chrifti  //  M.D..\.\.\  1 1 1 1 .'     (F,  1 20,  r.) 

Description.  8°,  10. 2  X  14.9  cm.,  the  text  being  6.-]  Y.  11.8 
cm.    1 19  ff.  unnumb.  +  i  blank  =  120  ff.,  31  11.    Niirnberg,  1534. 

Editions.  Rudolff  published  three  books  as  follows  :  ( i )  the 
Coss,  an  algebra,  in  1525  (see  p.  258  for  the  Stifel  edition  of 
1553);  (2)  the  Kiinstliche  Rechnung,  here  described;  (3)  a  col- 
lection of  problems  in  1530  (see  p.  159).  Of  the  Kunstliche 
Rechnung  the  following  editions  appeared  in  the  sixteenth  cen- 
tury :  Vienna  (Niirnberg  ?),  1526,  8°;   Niirnberg,  1532,  8°;   ib.. 


152 


RARA   ARITHMETICA 


1534  (here  described);  ib.,  1537,8°;  ib.,  1540  (below);  ib.,  1546; 
Niirnberg,  1553,  8°  (below);  ib.,  1557  (p.  153);  Vienna,  1561  ; 
Vienna  (Augsburg  ?),  1574;  Augsburg,  1588,  8°. 

This  work  is  an  extension  of  the  first  part  of  the  Coss,  and  is  divided 
into  three  parts:  (i)  Grundbiichlein,  the  fundamental  operations  with 
abstract  and  concrete  numbers,  integers,  and  fractions,  with  and  without 
the  abacus ;  (2)  Regelbiichlein,  the  rule  of  three  ('  Regel  de  Tri ')  and 
Welsch  practice  ('VVellisch  rechnung')  ;  (3)  Exempelblichlein,  problems 
and  results.  It  was  one  of  the  best-known  of  the  practical  arithmetics 
of  that  period.  The  rule  of  three  is  esteemed  highly  by  Rudolff,  for  he 
says :  '  sie  befchleufzt  in  fich  die  aller  niitzlichfte  Regel,  dadurch  unzeli- 
che  rechnung  in  kauffen  und  verkauffen  aufzgericht  werde.'  Of  the  Ital- 
ian method  of  solving  applied  problems,  the  '  Welsch  practice,'  he  says  : 
'  Dieweil  die  WeUifch  rechnung  nichts  anderes  ift,  daii  ein  gefchwinder 
aufzug  in  der  Regel  de  Tri  gegriindet,  wirt  fie  auch  derhalben  practica 
gefproche.' 

Other  luorks  0/1^26.  Ciruelo,  p.  60,  1495  j  Blasius,  p.  97,  1513; 
Tagliente,  p.  114,  1515;  Widman,  p.  37,  1489.  Sterner  mentions  an 
anonymous  Rechenbiichlein  as  printed  this  year  at  Niirnberg. 

CHRISTOFF    RUDOLFF. 

Ed.  pr.  1526.  Nurnberg,  1540. 

See  p.  151. 

Title.  This  is  substantially  the  same  as  in  the  1534  edition, 
but  bears  the  date  1540.    (F.  i,  r.) 

Colophon.  '  CGetruckt  zu  Niirmberg  bey  Johaii // Petreo/ Anno 
M.  D.  XL.'    (F.  117,  r.) 

Description.  8°,  8.8  X  13.9  cm.,  the  text  being  6.8  x  12  cm. 
117  ff.  unnumb.,  30  11.    Nurnberg,  1540. 

Editions.    See  above. 

CHRISTOFF   RUDOLFF.     Ed.  pr.  1526.     Nurnberg,  1553. 

Seep.  151. 
Title.  '  Kunftlich  rech-//nung  mit  der  ziffer  vnd  mit//den 
zal  pfenningen/ fampt  der // Wellifchen  Practica/ vnd  allerley// 
fortheyl  auff  die  Regel  //  De  Tri.//  Item  vergleichung  manch-// 
erley  Gewicht/ Elnmas/ Mfintz  ;c.  auff //etlich  Landt  vnd  Stett. 
//Gemehrt  mit  293  Exempeln/von  man-//cherleyKauffhendeln/ 


PRINTED    BOOKS  153 

mit  crklcrung/\vic//clic  Iclbcn  zu  machen  vnd  in  die // Rcgel 
zu  fetzcn  fcin.//Auff.s  new  widerumb  fleiffig  vberfehen/  //vnd 
an  vil  orten  gebeffert.// AUes  durch  Chriftoffen  Rudolff  zu // 
Wien  verfertiget.//  1553.'    (F.  i,  r.) 

Colophon.  '  Gedruckt  zu  Nurmberg/  durch  //  Gabriel  Hayn.// 
1553.'     (F.  206,  V.) 

Description.  8°,  10  X  157  cm.,  the  text  being  6.8  x  1 1.8  cm. 
206  ff.  unnumb.  +  2  blank  =  208  ff.,  24-26  11.    Nurnberg,  1553. 

Editions.    See  p.  152. 

As  the  title  states,  this  is  a  revision  of  the  1526  book,  with  some 
added  matter  and  a  considerable  number  of  new  examples.  I'he  new 
matter  begins  on  f.  T  8.  The  book  closes  with  a  list  of  gauger's  char- 
acters, 'die  vifier  ziffer,'  not  found  in  the  1534  edition,  the  integers 
being  represented  by  what  are  pracdcally  the  medieval  numerals,  and 
the  fractions  being  generally  unit  fractions. 

CHRISTOFF   RUDOLFF.      Ed.  pr.  1526.    Nurnberg,  1557. 

See  p.  151. 

Description.  This  edition  of  Rudolff' s  arithmetic  is  substan- 
tially verbatim  with  that  of  1553  (p.  152).  8°,  9.5  X  15  cm.,  the 
text  being  /x  1 1.8  cm.    206  ff.  unnumb.,  26  11.    Nurnberg,  1557. 

ANONYMOUS.     Ed.  pr.  1527.  Cologne,  1527. 

Title.  '  Compendia-//ria  artis  nvmerandi  ratio,  et  //  expeditif- 
fima  practicandi  uia,  figuris  Arithme//ticis  omnes  numeroru  for- 
mulas c6prehen//dens,  additis  quibufdam,  ut  raris,  ita//utilibus 
regulis.//  Radicis  Cvbicae  Extractio.// 

13886        151 
Cubus        653534     29952 


Radix  40  28 

I    2    I    2     o.  6 

96480 

38862144-'  (F-  I'  !■•) 

Colophon.    '  Coloniae  apvd    Melchiorem  //  Nouefienfem  Anno 
.M.  .D.  XXVII. //Menfe  maio.'     (F.  29,  r.) 


154  KARA   ARITHMETICA 

Description.  4°,  13.4X  19.4  cm.,  the  text  being  8.9  X  14.3 
cm.     29  ff.  unniimb.,  32  11.     Cologne,  1527. 

Editions.  There  was  no  other  edition.  The  dedicatory 
epistle  is  dated  '  Coloniae.  Anno  1527.  Calendis  Maijs.' 

This  extremely  rare  little  work,  almost  unknown  to  bibliographers, 
begins  with  a  theoretical  discussion  of  the  nature  of  number  and  arith- 
metic. This  is  followed  by  a  '  compendiaria  .  .  .  artis  numerandi  ratio 
in  tres  tractatus  digesta.'  Of  these  the  first  treats  of  the  '  species '  and 
the  rule  of  three  in  integers ;  the  second  of  fractions,  and  the  third  of 
business  problems.  The  work  is  too  theoretical  to  have  had  any  influence 
on  commercial  arithmetic. 

JOHANNES   WOLPHIUS.    Ed.  pr.  1527.     Frankfort,  1534. 

A  German  mathematician,  born  c.  1 500. 

Title.  '  Rvdi//menta  Arithmetices  //  Authore  lohanne 
Vuolphio  //  Herfbrugienfe.//  Elemen//tale  Geometricvm,  ex  // 
Euclidis  Geometria,  a  loanne  Voegelin,//  Haylpronnenfi,  ad 
omnium  Mathe//matices  studioforum  utili-//tatem  decerptum. 
//  Franc.  Chriftianus  Ege-//nolphus  excudebat.'  (The  title 
page  is  surrounded  by  an  elaborate  woodcut.     F.  i,  r.) 

Colophon.   At  the  end  is  the  date  'M.D.  XXXIIII.'  (F.  56,  r.) 

Description.  8°,  9.7  X  15  cm.,  the  text  being  6.8  X  11. 7  cm. 
56  ff.  unnumb.,  24-29  11.  (The  arithmetic  occupies  only  27  ff.) 
Frankfort,  1534. 

Editions.  Ntirnberg,  1527;  Frankfort,  1534,8°  (here  described); 
ib.,  1537;  Strasburg,  1539,  8°;  ib.,  1540;  Frankfort,  1548,  8° 
(below) ;  ib.,  1561. 

This  brief  treatise  on  arithmetic  covers  the  work  required  in  some 
of  the  Latin  schools  of  the  sixteenth  century.  It  contains  little  besides 
numeration  and  the  fundamental  operations,  including  duplation,  media- 
tion, the  rule  of  three,  and  fractions.  There  are  only  a  few  applications, 
coinage  and  partnership  being  the  most  prominent. 

JOHANNES   WOLPHIUS.     Ed.  pr.  1527.     Frankfort,  1548. 

See  above. 
Title.    '  Rvdi-//menta  Arithmeti-//ces,  Autore  loanne  Vuol-// 
phio     Herfbru-//gienfe.//  Elementa-//le    Geometricvm,//  Ex 


PRINTED    HOOKS  155 

Pluclidis  Geomctria,  a  loanne // VtKfjelin,  I  laylpronncnfi,  ad  o-// 

niniuni  Mathematiccs  ltudi()-/,'l"f>''^>'i^  utililatcm  (.\c-//vcvpium.// 

Franc.  Chri.  Ego.'     (F.  i,  r.) 

Colophon.    At  the  end  is  the  date  '  M.  D.  XLVIIL'    (F.  60,  r.) 
Description.    8°,  9.9  X  15.3  cm.,  the  text  being  8.2  x  1 1.8  cm. 

60  ff.  unnumb.,  22-27  ^'-     (The  arithmetic  occupies  only  28  ff.) 

Frankfort,  1548. 

FETRUS   APIANUS.     Ed.  pr.  1527.  Ingolstadt,  1527. 

Pf.tkr  Bien'EWItz,  or  Benxewitz.  Bom  at  Lei.snig,  in  1495;  ^^^^  '^'^ 
Ingolstadt,  April  21,  1552.  He  wrote  chiefly  on  astronomy,  and  in  his 
Cosmographia  (1524)  he  first  showed  how  to  determine  longitude  by  observ- 
ing the  distance  of  the  moon  from  certain  fixed  stars.  He  was  professor  of 
astronomy  at  Ingolstadt,  and  was  one  of  the  few  university  professors  of 
his  time  who  gave  instruction  in  arithmetic  in  the  German  language. 

Title.    See  Fig.  78. 

Colophon.  'Gedruckt  vnd  volendt  zu  Ingolftadt //durch  Ge- 
orgium  Apianum  von  Leyi3-//nick/ jm  Jar  nach  der  geburt  Chrifti 
//  1527.  am  9.  tag  Augufti.'     (F.  299,  v.) 

Description.  8°,  10  X  14.8  cm.,  the  text  being  7  X  11. 7  cm. 
299  ff.  unnumb.  +  i  blank  =  300  ff.,  27  11.     Ingolstadt,  1527. 

Editions.  Ingoldstadt,  1527,  8°  (here  described);  Frankfort, 
1537,  8°  (p.  157);  ib.,  1544,  8°;  ib.,  1564,8°;  ib.,  1580.  Graesse 
mentions  an  '  Arithmetica,'  Leipzig,  1543,  8°,  but  questions  it, 
and  Romstock  does  not  give  it  in  his  article  on  Apianus  in  the 
Astronomen,  Matheniatikcr,  iind  Physikcr  der  Diocese  Eichstdtt. 

Apianus  follows  Rudolff  so  closely  as  to  give  ground  for  comment. 
His  arithmetic  differs  from  the  latter's  chiefly  in  the  arrangement  of  the 
matter.  The  work  is  largely  commercial,  and  includes  the  fundamental 
operations  and  the  ordinary  rules  and  applications  of  the  period.  There 
is  a  chapter  on  counters  at  the  end  of  the  book.  Indeed,  Apianus  advises 
their  use,  saying  :  '  die  Sumering  der  Regifter  durch  die  rechenpfening 
auff  der  lini  brauchfamer  ift  dan  durch  die  federn  oder  kreide.'  The 
work  is  also  interesting  on  account  of  its  quaint  illustrations.  The 
title  page  (p.  156)  is  noteworthy  on  account  of  the  engraved  'Pascal 
triangle'  a  century  before  Pascal  studied  this  numerical  form,  and 
some  years  before  Stifel  mentioned  it,  and  because  of  the  picture  of  line 
reckoning.     I  know  of  no  example  of  the  '  Pascal  triangle  '  in  print 


56 


RARA   ARITHMETICA 


before  this   one,  although  the  arrangement  had  doubtless  long  been 
more  or  less  familiar  to  mathematicians. 


I  nuitor  in  x>it^m  0achcnt/mic  fchoncrt  ^c 

I  gelrf vn  fragfliicFcit  Gectnffcii  ♦  t^urtOcr^ 

lid)  vcas  fcttl  vnnty  Sc$>enX)i^i^Ait  in  t)cr 

^tfit^c/  tjesglcvdxrt  furitirtlf?  witjer  in 
^eif^fd)cv  rtctf)  in  Wclfd^tr  fpi<id)nk 
get>iiicPc.  ^iirch  J3ctrum  Zpiai 
"nicF/^2t|lrortomci 
joljlrtt  <2)rt)iit4«' 
riu/vcrfcmgcc. 


ns?  fT»T* 


Fig.  78.   Title  page  of  the  first  edition  of  apianus 

0///er  works  0/1527.  FeHciano,  p.  146,  1526  ;  Riese,  p.  139,  1522, 
the  first  Niirnberg  edition  so  far  as  I  know;  Tagliente,  p.  114,  1515  ! 
W.  Peer,  '  Ain  new  guet  Rechenbuchlein,'  Niirnberg,  8°. 


PRINTED    HOOKS  157 

PETRUS   APIANUS.     Ed.  pr.  1527.  Frankfort,  1537. 

See  p.  155. 
Title.  '  Ein  ncvve  vnd  wolgcV/griindte  vndcrweifung  aller // 
Kauffmans  Rcchnung  in  dreien  Bu//chern/  mit  fchonen  Regeln 
vnd  fragftucken  bc-//griffcn.  Sunderlich  was  fortel  vnnd  bchen- 
digV/keit  in  der  Welfchcn  Practica  vnnd  Tolle-//ten  gebraucht 
wurt/des  gleichen  vor//mals  wcdcr  inn  Tcutfcher  noch  in// 
Welifchcr  Spraach  nic  getruckt.//  CDurch  Petrum  Apianum 
von  Leyfznick  der // Aftronomei  zu  Ingolftatt  Ordinarium.// 
(Woodcut  of  merchants  using  counters.)  Franc.  Chri.  Egen.' 
(F.  I,  r.) 

Colophon.  '  Zu  Franckfurt/ bei  Chriftian  Egenolff/ //Anno 
Domini.  M.  D.  x.xxvij.//Im  Herbftmon.'     (F.  183,  v.) 

Description.  8°,  9.8  X  15.3  cm.,  the  text  being  7  x  11.5  cm. 
183  ff.  unnumb.  +  i  blank  =  184  ff.,  25-28  11.    Frankfort,  1537. 

See  p.  155. 

JOHANNES   FERNELIUS.     Ed.  pr.  1528.  Paris,  1528. 

Jean  Kernel.  Born  at  Clermont  in  1497;  died  at  Paris,  April  26,  1558. 
He  was  a  physician,  with  a  taste  for  mathematics  and  astronomy.  He 
wrote  numerous  works  on  medicine  and  mathematics. 

Title.  '  loannis  Fer//nelii  Ambianatis  //  de  proportionibus 
Libri  duo. //Prior,  qui  de  fimplici  proportio-//ne  eft,&magnitudi- 
num  &  nu-//nierorum  turn  fimplicium  tum  //  fractorum  rationes 
edocet.//Pofterior,  ipfas  proportiones  co-//parat :  earumq3  rati- 
ones colligit.// Parisiis // Ex  aedibus  Simonis  Colinaei // 1528.' 
(F.  i,r.    Fig.  78.) 

Colophon.  'CLibellorvm  de  proportionibvs,  loanne  //  Fernelio 
Ambianate  authore,  finis.'     (F.  28,  v.) 

Description.  Fol.,  22.2  X  31.5  cm.,  the  text  being  15.6  X  24.4 
cm.     4  ff.  unnumb.  +  24  numb.  =  28  ff.,  44-45  H.     Paris,  1528. 

Editions.    There  was  no  other  edition. 

This  is  one  of  the  best  of  the  sixteenth-century  treatises  on  the 
mediaeval  proportion.  It  follows  the  Boethian  treatment,  as  seen  also 
in  the  work  of  Bradwardin. 


158 


RARA  ARITHMETICA 


Fig.  79.   Title  page  of  fernelius 


PRINTED    BOOKS  159 

Othenvorks 0/ /j28.  Borghi,  p.  i6,  1484;  Cassiodorus,  p.  j  i  i,  1540; 
Ciruelo,  p.  60,  1495  ;  Riese,  p.  139,  1522;  TaglieiUe,  p.  114,  1515  ; 
Christiernus  Torchillus  Morsianus,  'Arithmetica  brcvis  et  dilucida  in 
quinque  partes  digesta,'  Cologne,  8°  (but  see  p.  182,  1536). 

Works  of  i^2g.  Anianus,  p.  32,  1488;  Riese,  p.  139,  1522  ;  Tonstall, 
p.  135,  1522  ;  B?eda,  '  De  natura  rerum  et  temporum  ratione  libri  duo,' 
Basel,  cap.  I  being  '  De  cominitu  vel  loquela  digitorum';  there  was  an 
edition  by  Noviomagus,  Cologne,  1537  (see  also  pj).  131,  140,263,  1521, 
1525,  1554)- 

1530- 


CHRISTOFF   RUDOLFF. 

Ed.  pr. 

1530. 

Augsbur; 

Seep.  151. 

Title.    Sec  Fi^.  80. 

bey  /  em  iiU6li*e  ^f  ttfiruction/  tr<5Ut^ 

cber  %t^ii\^t  Oic  verglcychrius  /  Ocr  ^Inma^ 

t»nrcb  t»cit  Strcf  el/bcr  pfunO  t>urct)  <x\>xct^ 

gen/  tcr  2:^.Mvt>/  tt>eyrt/  »nb  (Dima|i:c* 

2)urch  abfcdbtcrtrt  t>er  tllorf^'  tiird) 

grtttgt;(tfftert  yrcrt  tvcrOc  in,t^zn  tixi 

(kmtx  sSerlcrrtcit  t>n  ergKineefcy* 

gw  tX'ien  m(Df!crrcyi/^urc^ 

<;«d)  alien  /^^n^c^ic^ 

rwrtgcn  pcrfbtten 

56riUQt)ttbgd# 

tctn  t)er# 


M.    a   XXX, 

Fk;.  80.    Title  pace  oe   ihe  first  edition   ok  kudolfk's  Exempel 
B  lie  hi  in 


i6o  RARA   ARITHMETICA 

ColopJion.  'Getruckt  in  der  loblichen  Reychftat  Aiig-//fpurg/ 
durch  Heynrichen  Stayner/  // Volendet  am  3 1  May  im  jar/7M.  D. 
XXX.'    (F.  75,  V.) 

Description.  8°,  9.9  X  15.1  cm.,  the  text  being  7.4  x  12.2  cm. 
75  ff.  unnumb.  +  2  blank  =  yj  ff.,  28  11.     Augsburg,  1530. 

Editions.  Augsburg,  1530,  8°  (here  described);  Niirnberg, 
1538;  ib.,  1540. 

This  is  the  third  of  Rudolff's  works  (see  p.  15 1).  As  the  name  imphes, 
it  is  merely  a  collection  of  problems,  two  hundred  and  ninety-two  in 
number.  Most  of  these  problems  are  of  a  genuine  business  nature,  and 
they  furnish  a  good  idea  of  the  ordinary  commercial  needs  of  the  first 
half  of  the  sixteenth  century  in  Germany. 

ORONTIUS   FINAEUS.     Ed.  pr.  1530-32.    Paris,  1530-32. 

Oronce  Fin6.  Bom  at  Brian9on  in  1494  ;  died  at  Paris,  October  6,  1555. 
He  was  made  professor  of  mathematics  in  the  (later  called)  College  de 
France  in  1532.  He  wrote  extensively  on  astronomy  and  geometry,  but 
was  not  a  genuine  scholar. 

Title.    See  Fig.  81. 

Colophon.  '  Excvsvm  est  avtem  ipsvm  opvs  PaZ/rifijs  in  uico 
Sorbonico,  impenfis  Gerardi  Morrhij,  &  loannis  Petri.  Anno  // 
M,D.XXXII.'     (F.  216,  r.) 

Description.  Fol.,  24.1  X  37  cm.,  the  text  being  18.4  x  27.7 
cm.    9  ff.  unnumb. +  208  numb.  =  217  ff.,  48  11.    Paris,  1530-32. 

The  title  page  of  the  geometry  appears  on  f.  49,  r.,  with  the 
date  M.  D.  XXX;  the  cosmography  on  f.  lOi,  r.,  with  the  date 
M.  D.  XXX  ;  the  horography  on  f.  157,  r.,  with  the  date  M.  D. 
XXXI. 

Editions.  Paris,  1530-32,  fol.  (here  described) ;  ib.,  1535;  ib., 
1542,  fol.  (p.  163);  ib.,  1544,  8°  (p.  163);  ib.,  1554;  ib.,  1555^ 
4°  (p.  163);  Venice,  1587,  4°  (p.  164).  Leslie's  statement  that 
the  work  appeared  in  1525  is  unfounded.  For  the  '  De  rebus 
mathematicis,'  1556,  see  p.  279. 

This  is  the  first  edition  of  the  works  of  Finaeus,  perhaps  the  most  pre- 
tentious French  mathematician  of  his  time,  and  was  published  during 
the  years  1530-32.  The  dedicatory  epistle  is  dated  '  Lutetije  Parifiorum 
Calendis  lanuarij  1531,'  or  1532  new  style.    The  part  on  arithmetic  is 


PRINTED    HOOKS 


i6r 


Fig.  8i.    I'itle  page  of  the  first  edition  of  finaeus 


l62 


RARA   ARITHMETICA 


divided  into  four  books  dealing  respectively  with  integers,  common  frac- 
tions, sexagesimal  fractions,  and  proportion.    There  are  no  applications 


ORONTII  FINEI  DELPH. 

JO  primi  elemeinorum  Euclidis  facile  manifeftatur .  Sc  angujus  A  B  M  ,  an* 
oulo  A  C  F  eft  ajqualis  (  nam  utercp  re£his)  igitur  per  4  fexti  aufdcm  Eudidis, 
fit  ficut  H  B  ad  B  A,  ica  F  C  putei  laacudoad  G  A  tompoCtam  ex  G  B  &  B  A  longiV 
tuHinem.fiue  profunditatrm. 

Sit  exempli  gra"tia  B  H  2o  partium.qualium  latus  quadrati  eft  6o;b  E  aute  nie<= 
tiatur,8i  fit  in  exemplum  6  cubicorum.tot  etiam  cubirorum  erit  G  F:funt  enim  la 
tcra  perallclogrammi  B  E  F  C  oppofita.quje  per  J4  eiufdem  primi  funt  inuicera 
sequalia.Duc  igitur  6  in  6ofiem  j6o:qus  diuidepcr  10,  &.habd>is  pro  quodee 
te  iS.Tot  igitur  cubitore  erit  A  c: 
a  quafi  dempferis  A  B  trium  uer^ 
bi  gratia  cubiiorum,  relinquptur 
B  C  dcfyderata  &  in  profundum 
depfla  putei  logiiudoiycubicoFii, 
IDEM  Q_V  O  Q.V  E  SIC  O  B» 
tinebis.  Metire  H  e:  fitcp  exempli 
caufa  y  cubicoru.  Deinde  multi* 
plica  y  per  6o.fient  joo:hapc  diui 
per  iO.produccntur  ly.uelut  an» 
tca.Bina nancj)  triaiigula  A  BH  et 
H  E  F  funt  rurfum  a?quiangula.  (jc§ 
quoniam  angulus  A  h  B  angulo 
E  H  F  ad  uertjcem  pofito ,  per  15 
primi  Euclidis  eft  aequalis.ite  re< 
<fVus  qui  ad  B,  recto  qui  ad  E  pari 
ter  aequat.  reliquus  igitur  bah 
reliquoH  F  E  per  ^2  eiufdem pnV 
micftsqoalis.  Vnde  per  fupee 
rius  allegata  quarta  propofitione 
fexti.ticut  HE  ad  B  A.ica  H  E  ad  E  F,eidcm  B  G  per  hypothelTm  sequalem. 

Cum  autem  acciderit  putcum  rotundam  habere  figuram.habenda  erit  cofydts 
ratio  diametri  putealis  orificn,&  reL'qua  omnia  uelutiprius  abfoluenda, 

<j^  RELIQ.VVMETS,  VT  ' 

eandem  rerum  in  profunditdes 
preflfarum,  per  uulgatu  quadras 
tern  metiri  doceamus  altirudine. 
Sit  itacp  putcus  circularis  E  F  G 
H.cuius  diamcrer  lit  E  F.aut  illi 
aequaiisG  H.Adplica  igitur  qua- 
dratem  ipfi  putei  orificio:  in  hijc 
modu.ut  finis  lateris  A  D  ad  datii 
puncflum  E  conftituatur .  Leua 
poftmodu,autdeprime  quadra^ 
tem(libero  femper  dcmiflb  per- 
pendiculo)donec  radius  uifuaiis 
per  ambo  foramina  pinnacidioRt 
ad  inferiorem  &  e  diametrofi- 
:  term  inSijperducat.  Quo 


s.m.u  .crminu  H  perducat.Quo        ^^^ 

fafto&immoto  quadrate,  uide    ^^-^ 

in  qua  ^^  -^-" 

Fig.  82.    From  the  first  edition  of  finaeus 


worthy  the  name,  and  the  work  has  little  to  commend  it.  Some  inter- 
esting illustrations  showing  the  use  of  the  mediaeval  Quadrans  are  shown 
in  Fig.  82. 


PRINTia)    BOOKS  163 

ORONTIUS   FINAEUS.    Ed.  pr.  1 530-32.  Paris.  1542. 

See  p.  160. 

Title.  '  Orontii  //  Finci  Dclphin.  Re-//gii  Matheniaticarvm  // 
Professoris: //arithmetica // practica,  libris  qva-//tu()r  abfoluta, 
omnibus  qui  Ma-//thematicas  ipfas  tractare  volunt  //  perutilis, 
admodumqueneceffa-//ria:  Ex  nouiffima  authoris  reco-//gnitione, 
amplior,  ac  emenda-//tior  facta.// yEdito  tertia.// Parisiis.// Ex 
officinaSimonisColinaei.//  1542. //Cum  gratia&priuilcgioChri-// 
ftianifiimi  P>ancorum  Regis.'     (V .  i,  r.) 

Description.  Fol.,  20.9  X  30.  i  cm.,  the  te.xt  being  16  X  28.2 
cm.     2  ff.  unnumb.  +  66  numb.  =68  ff.,  40  11.    Paris,  1542. 

See  p.  160. 

ORONTIUS   FINAEUS.     Ed.  pr.  1530-32.  Paris,  1544. 

See  p.  160. 

Title.  '  Orontii  //  Finaei  Delphi-//natis,  Regij  Mathe-//niati- 
caru  Lutetise // Profefforis,// Arithmetica // Practica,  in  com- 
pendiu  per  Authorem  //  ipfum  redacta,  multifqj  acceffionibus 
//locupletata:  Ijs  qui  ad  liberam  quauis,// nedu  Mathematica 
adfpirant  philofo-//phia  perutilis,  admodiimqj  neceffaria.//  Lvte- 
tiae  Parisiorvm // Apud  Simonem  Colinaeum.// 1544.// Virefcit 
vulnere  virtus.'    (F.  i,  r.) 

Description.  8°,  11  X  17  cm.,  the  text  being  8.6  x  12.8  cm. 
95  ff.  numb.  +  I  unnumb.  =96  ff.,  31  11.     Paris,  1544. 

See  p.  160. 

ORONTIUS   FINAEUS.     Ed.  pr.  1530-32.  Paris,  1555. 

See  p.  160. 

Title.  'Orontii  Finaei //Delphinatis,  Regii// Mathematicarum 
Lutetiae //profefforis, //de  arithmetica  practi-//ca  libri  quator: 
Ab  ipfo  authore  uigi-//l^nter  recogniti,  multifque//accefsionibus 
recens // locupletati.// Lvtetiae  Parisiorvm, //apud  Michaelem 
Vafcofanum,//  1555.// Ex  privilegio  regis.// Virefcit  uulnere 
uirtus.'    (F.  I,  r.) 

Description.  4°,  15.4  X  20.7  cm.,  the  text  being  i  1.6  X  17.2 
cm.    4  ff.  unnumb.  +  72  numb.  =  76  ff.,  34-35  11.     Paris.  1555. 


i64  RARA   ARITHMETICA 

ORONTIUS   FINAEUS.     Ed.  pr.  1530-32.       Venice,  1587. 

See  p.  160. 

Title.  'Opere//di//Orontio  Fineo//del  Delfinato://Diuife 
in  cinque  Parti  ;//Arimetica,  Geometria,  Cofmografia,  &  Oriuoli, 
//  Tradotte  //  Da  Cofimo  Bartoli,  Gentilhuome,  &  Academico 
Fiorentino  : // Et  gli  Specchi,// Tradotti  dal  Caualier  Ercole 
Bottrigaro,  Gentilhuomo  Bolognefe.//Nuouamente  pofte  in  luce: 
//con  privilegio. // In  Venetia,  Preffo  Francefco  Francefchi 
Senefe,  1587.'     (F.  i,  r.) 

Description.  4°,  14.9  X  20.9  cm.,  the  text  being  10.  i  X  16.8 
cm.  8  ff.  unnumb.  +  81  numb.  =  89  ff.  (in  the  part  devoted  to 
arithmetic),  35-39  11-    Venice,  1587. 

Editions.  See  p.  160.  That  this  is  the  first  Itahan  edition 
appears  in  the  printer's  dedicatory  epistle  to  Guidubaldo  de' 
Marchesi  del  Monte,  in  which  he  mentions  '  in  quefta  occafione 
deir  hauere  ftampato  I'opera  d'Orontio  nella  noftra  Tofcana 
lingua,'  with  the  date  '  Di  Venetia,  il  di  7.  di  Luglio,   1587.' 

See  p.  160. 

ANDREAS   ALCIATUS.     Ed.  pr.  1530.         Hagenau,  1530. 

Bom  at  Alzano,  near  Milan,  May  8,  1492;  died  there  June  12,  1550.     He 
was  an  Italian  jurist. 

Title.    See  Fig.  83. 

Description.  8°,  10.3  X  15.4  cm.,  the  text  being  7.8  X  ii.i 
cm.    50  ff.  and  2  plates  unnumb.,  28  11.     Hagenau,  1530. 

Editions.  There  was  no  other  separate  edition,  but  the  works 
of  Alciatus  appeared  at  Basel  in  1571,  3  vol.,  fol. 

The  work  of  Alciatus  extends  only  to  f.  D  i.  The  'Oratio  de  legibus' 
of  Melanchthon  then  extends  to  f.  F5.  This  is  followed  by  '  Budsei 
qvEedam  de  moneta  Graeca,'  etc.  The  work  of  Alciatus  is  not  an  arith- 
metic, but  a  history  of  weights  and  measures.  As  such  it  is  of  value  for 
the  historical  development  of  commercial  mathematics. 

Other  works  0/1530.  Boethius,  p.  27,  1488;  Bradwardin,  p.  61, 
1495;  Clatovenus,  p.  292,  1558;  Maffei,  p.  86,  1506;  Riese,  p.  139, 
1522;  Tagliente,  p.  114,  1515;  Torrentini,  p.  76,  1501  ;  Johann 
Kolross,  a  primer  entitled  '  Enchiridion  :  das  ist  Handbiichlin  tiitscher 


PRINTED    BOOKS  165 

Orthographi  .  .  .  Auch  wie  mann  die  Cifer  vnd  tiidtsche  zaal  verston 
sol,'  Basel,  with  another  edition,  ib.,  1534,8°.  There  was  also  an 
anonymous  work  entitle<l  '  La  vraye  maniere  pour  apprendre  a  chiffrer 
et  compter,'  published  at  Lyons,  s.  a.,  12°,  c.  1530. 

AND^  AL 

CIATI    LIBELLVS, 

DE    PONDERlBVrS   ET 

menfuris  • 

ITEM 

Budxt  ft£(itm  de  eadem  re,4dhuc  non  mfi  * 

ITEM 

Phtlippi  h\eUnchthomu  de  ijfdemM 

Garmnorum  ufum,fcntenU4- 

AkUtt  quoc^.O"  Mippi  Mehnchthontf, 

in  laitdm  luru  Cirnlts ,  oratmcs  du£ 

€lfg4nupm<e* 


Hagano£  apud  loUtt'  Sec^ 
Anno  M.  D.  XXX» 

Fk;.  83.    Title  page  of  alciatus 

JOACHIM   FORTIUS   RINGELBERGIUS. 

Ed.  pr.  1 53 1.  Lcydcn,  1531. 

Joachim  Sterck  Ringelbergh.  Born  at  Antweq?,  c.  1499;  ^^^^^  ^• 
1536.  He  taught  philosophy  and  mathematics  in  various  cities  of  Germany 
and  France. 

Ttt/e.  Toachimi  //  Fortii  Ringel-//bergij  Andouerpiani  opera, 
//quae  proxima  pagina//enumerantur.// Virtvte  dvce  //comite 
Fortvna.//Apvd  Gryphivm  //  Lvgdvni,  //anno//M.  D.  XXXI.' 
(P.  I.) 


i66  RARA   ARITHMETICA 

Colophon.  '  Lvgdvni  apvd  //  Seb.  Grypivm,//  anno  //  M.  D. 
XXXI.'     (P.  687.) 

Description.  8°,  10. 3  X  15.7  cm.,  the  text  being  8.4  X  12.2 
cm.     687  pp.  numb. +  30  =  717  pp.,  22-29  11.    Leyden,   1531. 

16     I  O.    F  O  R  T  I  I     RING. 

V-^YAmiim  tiwnm  hoc  pd^o  digmmtur, 

'^      3 


Omnii 

Fig.  84.    From  thk  first  edition  ok  ringelberoius 

Editions.    Leyden,  1531,  8°  (here  described);  ib.  (at  least  the 
arithmetic  part),  1539,8°;  Basel,  1541,  8°  (see  p.  167);   Leyden, 


PRINIKI)    HOOKS  167 

1556,  8°  (see  p.    168).    The   '  Epistola   ad   Lcclorem  '    is   dated 
'Louanij  Idib.  Augusti,  Anno  M.  D.  XXIX.' 

This  work  of  Ringelhergius  is  somewhat  encyclopedic  in  charac- 
ter. The  'Liber  de  Ratione  ftiidij,'  with  ' Annotationes'  thereon  and  a 
'Horoscopus  Hbri  ratione  ftuchj '  to  show  that  it  was  written  in  an  aiis]>i- 
cious  time,  is  followed  by  six  other  books.  These  relate  to  (Grammar, 
Dialectics,  Rhetoric,  Mathematics,  and  Divination,  closing  with  a  book 
entitled  '  Commvnis  cvivsdam  Naturaj  funt.'  The  book  on  mathematics 
includes  a  chapter  on  arithmetic  in  which,  in  17  ])ages  (about  10  pages 
excluding  the  illustrations),  the  author  treats  of  the  Boethian  ratios, 
figurate  numbers  (see  Fig.  84),  and  the  fundamental  operations  with 
figures  and  upon  the  line  abacus.  The  part  relating  to  astronomy  had 
already  been  published  at  Basel  in  1528,  and  the  cosmography  in  Paris 
in  1529. 

Other  7V07'ks  of  ijjT.  Kobel,  p.  10 1 ,  15  14  ;  Juan  (}uti<^rrez  de  (iualda, 
'  Arte  breue  y  muy  prouechos  de  cuenta  castellana  y  arismetica,'  Toledo, 
4°;  ib.,  1539;  Saragossa,  1557,  1564;  Alcala,  1570  ;  H.  C.  Agrippa, '  De 
occulta  philosophia  libri  tres,'  s.  1.,  with  later  editions,  Cologne,  1533, 
fob;  1541  ;  Lugduni,  1550,  8°;  s.  1.,  1565;  Basel,  1567,  8°;  s.  1.  a. 
(Paris,  1567  ?),  included  by  De  Morgan  without  much  reason. 

JOACHIM    FORTIUS   RINGELBERGIUS. 

Ed.  pr.  1 53 1.  Basel,  1541. 

See  p.  165. 

Title.  '  Ioachimi//Fortii  Ringelber//gii  Andoverpiani  Ivcvbra- 
//tiones,  uel  potius  abfolutifsima  /cu/cXoTrai'StS^ :  nem-//pe  liber 
de  Ratione  ftudij,  utriusq;  linguae,  Gramatice,//  Dialectice,  Rhe- 
torice,  Mathematice,  &  fublimioris  //  Philofophiae  multa.  Quoru 
eXeyxo^i  fub  fecjuenti  pa//gina  enumeratur.  Atq;  haec  omnia  eo 
iudicio  &  //ordine  funt  tradita,  ut  uel  fola  cuiq;  //mcliorum  lite- 
rarum  ftudiofo // fatis  ad  fummum  inge-//nij  cultum  effe//pof- 
sint.//Basileae.//Anno  M.  D.  XLI.'    (P.  i.) 

Colophcm.  *  Basileae  apvd  Bartholo-//mevm  Vvesthemervm  // 
anno  M.  D.  XLI.'    (P.  797.) 

Description.  8°,  10.3  X  15.7  cm.,  the  text  being  8.2  X  11. 8 
cm.     796  pp.  numb.  +  2  unnumb.  =  798  pp.,  25  11.    Basel,  1541. 

See  p.  166.  Although  the  title  differs  from  that  of  1531,  the  work  is 
the  same. 


i68  RARA   ARITHMETICA 

JOACHIM   FORTIUS   RINGELBERGIUS. 

Ed.  pr.  1 53 1.  Leyden,  1556. 

See  p.  165. 

Title.  '  loachimi  //  Fortii  //  Rin-//gelbergii  //  Andoverpiani 
// Opera,// Quae  proxima  pagina  enumerantur.// Lvgdvni,// 
Apud  loannem  Frellonium.'  (P.  i.  The  rest  of  the  page  is 
torn  off.) 

Colophon.  '  Lvgdvni,// ex  officina  typogra-//phica  Michaelis  // 
Sylvii,//M.  D.  LVL'     (P.  663.) 

Description.  8°,  10. 5  X  15.2  cm.,  the  text  being  9X  13  cm. 
4  pp.  unnumb.  +  4  blank  +  660  numb.  =  668  pp.,  26-31  11. 
Leyden,  1556. 

See  p.  166.  Although  the  title  differs  from  that  of  153 1,  the  work  is 
the  same. 

MICHAEL   PSELLUS.     Ed.  pr.  1532.  Venice,  1532. 

Called  the  Younger,  to  distinguish  him  from  a  philosopher  of  the  same 
name  who  lived  about  870  a.d.  Born  at  Constantinople  in  1020;  died  in  a 
cloister  in  mo.  He  studied  at  Athens  and  taught  philosophy  at  Con- 
stantinople. 

Title.    See  Fig.  85. 

Description.  8°,  9.9  X  14.9  cm.,  the  text  being  7.4  X  1 1.4  cm. 
I  f.  blank+  104  unnumb.  =  105  ff.,  24-25  11.   Greek,  Venice,  1532. 

Editions.  Venice,  15  32, 8°,  Greek  (here  described);  Paris,  1538, 
Greek,  4°;  Paris,  1545,  Greek  and  Latin;  Augsburg,  1554,  8°, 
Greek  and  Latin;  Wittenberg,  1556,  Latin;  Basel,  1554  and 
1556,  8°,  Greek  and  Latin;  Paris,  1557,  8°,  Latin  (p.  170); 
Wittenberg,  1560,  Latin;  Paris,  1585,  Latin;  Leipzig,  1590, 
8°,  Greek  and  Latin  (p.  170);  Heidelberg,  1591,  Latin;  Tours, 
1592,  Latin. 

Psellus  was  one  of  the  last  of  the  Greek  writers  on  arithmetic.  This 
part  of  his  work  is  devoted  solely  to  the  theory  of  numbers,  and  it 
represents  the  arithmetical  inheritance  derived  from  the  older  Hellenic 
civihzation.  The  treatise  covers  the  mediaeval  Quadrivium — arithmetic, 
music,  geometry,  and  astronomy  —  and  is  the  only  late  Greek  work  on 
arithmetic  that  attracted  attention  in  the  Renaissance  period.  The  arith- 
metic is  merely  a  primer  for  the  study  of  Nicomachus. 


PRINTED    HOOKS  169 

Other  works  of  1532.  Aventinus,  p.  136,  1522;  Capella,  p.  66,  1499  ; 
Feliciano,  p.  146,  1526;  Kobd,  p.  102,  1514;  Riulolff,  p.  151,  1526; 
Stifel,  p.  223,  1544;  Vincenlo,  p.  140,  1522  ;  Johann  Brandt,  'Kunst- 

Ad^omg  ^'(7i;Vo7rf 0?  As^ovo/u/ocj . 

SAPIENTISSIMI  PSEL 

li  opus  dilucidum  m  quattuor  Ma/ 

thematicas  difciplinas,  Arith/ 

mcticam  ,Mu  ficam  ,Geome 

triam,K  Aftronomiam. 

Kutncrornm  hie  contraftior  explication 
Elaboratum  Mufices  Compendium* 
Copendiu  rurfiis  Geometric  rationiU 
Aftronomij  coaftio  perfpicua* 

VENETIJS.  MDXXXIL 

CCumgratia^ 

Fic.  85.   Title  page  ok  the  first  edition  of  psellus 

liche  Rechnung  mit  der  Zyffern  \Tid  Pfennigen,  Auff  allerlcy  handt- 
tierung,'  Cologne,  8°,  39  ff. ;  Georg  Reichelstain,  '  Kauffmans  hand- 
biichlin.  Aller  Rechennschafft  behendigkeyt,  auff  Linien  vnd  Ziffern,' 
Frankfort,  sm.  8°,  with  another  edition  in  1534. 


I70  RARA   ARITHMETICA 

MICHAEL   PSELLUS.     Ed.  pr.  1532.  Paris,  1557. 

See  p.  168. 

Title.  '  Michael  //  Psellvs  de  //  Arithmetica,//  Mvsica,  Geo- 
metria://&  Proclus  de // Sphaera,// Elia  Vineto  Santone  inter- 
prete.//  (Woodcut  with  motto :  Inpingvi  Galhna.)  Parisiis,// 
Apud  Guhelmum  Cauellat,  in  pingui  ganina,//ex  aduerfo  collegij 
Came racenfis./V  1557.'     (F.  i,  r.) 

Description.  8°,  10.6  X  16.2  cm.,  the  text  being  6.6  y.  13.3 
cm.  2  ff.  unnumb.  +  76  numb.  +  3  blank  =  81  ff.  (18  ff.  on 
arithmetic),  22  11.     Paris,  1557. 

Editions.    See  p.  168.     This  is  one  of  the  Latin  editions. 

See  p.  168. 

MICHAEL   PSELLUS.     Ed.  pr.  1532.  Leipzig,  1590. 

See  p.  168. 

Title.  '  PselU  //  Philofophi  &  Mathemati-//ci  clarifsimi  // 
Arithmetica// Editaftudio//M.  Chriftophori  Meureri,//Mathe- 
matum  Prof eff oris  publici//in  Academia // Lipfienfi.//  1590.// 
Plato  interrogatus,  cur  homo  fit  a-//nimal  fapientifsimum  :  o-n 
apiB-l lix.dv  iiriqaTai,  refpondit.// Lipsiae.'    (F.  i,  r.) 

Colophon.  '  Lipfiae,  imprimebat  Michael  //  Lantzenberger.// 
Anno  M.  D.  XC    (F.  24,  v.) 

Description.  8°,  9.5  X  15.8  cm.,  the  text  being  6.6  X  12.3  cm. 
24  ff.  unnumb.,  30  11.     Leipzig,  1590. 

Editions.  See  p.  168.  The  Latin  dedicatory  epistle  is  dated 
'Lipfiae  XV.  Cal.  Nouemb.  Anno  poft  Chriftum  natum  1590.' 
The  text  is  in  Greek  and  Latin. 

See  p.  168. 

MICHAEL   PSELLUS.     Ed.  pr.  1532.  Leipzig,  1616. 

See  p.  16S. 
Title.    '  Pselli  //  arithmetica  //  Guilhelmo    Xylandro  //  inter- 
prete  //  Cum   Praefatione  //  Christophori   Meureri  D.//  Mathe- 
matum  Profefforis  //in  Academia  Lipfienfi.//  1616.//  Lipsiae// 
Typis  Abrahami  Lambergi.'     (F.  i,  r.) 


PRINTKH    HOOKS  171 

Description.    8°,  9  x  14.4  cm.,  the  text  bcin^'  6.6  x  12.1  cm. 
39  pp.,  1-8  unnumb.,  then  numb.  1-31;  24-25  11.    Leipzig,  1616. 
See  p.  168. 

ADAM    RIESE.     Ed.  pr.  1533.  Leipzig,  1 536. 

See  p.  ijcS. 

Title.    See  Fig.  86. 

Descnption.  4°,  14.6  X  19  cm.,  the  text  being  9.6  X  15.1  cm., 
and  the  tables  9.5  X  12  cm.  79  ff.  unnumb.  +  i  blank  =  80  ff., 
17-24  11.  in  the  Introduction  (3  ff.),  the  rest  of  the  book  con- 
sisting of  tables.     Leipzig,  1536. 

Editions.    Leipzig,  1533;   ib.,  1536,  4°  (here  described). 

This  is  a  set  of  mercantile  tables  for  the  multi])licalion  and  division 
of  denominate  numbers. 

GEORGE   AGRICOLA.     Ed.  pr.  1533.  Paris,  1533. 

Bom  March  24,  1490,  at  Glauchau,  Saxony;  died  November  21,  1555,  at 
Chemnitz.  He  was  rector  of  a  school  at  Zwickau  (i  518-1522),  and  in  later 
life  a  physician.    He  wrote  a  number  of  scientific  works. 

Title.    See  Fig.  87. 

Descnption.  8°,  10. 2  x  15  cm.,  the  text  being  6.8  X  12.4  cm. 
7  pp.  unnumb.  +  3-261  numb.  +6  blank  +  i  with  woodcut  =  273 
pp.,  28  11.     Paris,  1533. 

Editions.  Paris,  1533,  8°  (here  described);  Venice,  1533,  8°; 
ib.,  1535,  fol.;  Basel,  1549,  8°;  ib.,  1550,  fol.  There  was  also 
an 'Epitome  omnium  GeorgiiAgricolae  de  mensuris  et  ponderibus 
per  G.  Philandrum'  published  at  Lyons  in  1552,  8°. 

The  work  can  hardly  be  called  an  arithmetic,  but,  like  a  few  others 
included  in  this  list,  it  is  a  valuable  book  of  reference  on  the  history  of 
ancient  measures.  It  consists  of  five  books  as  follows :  '  Liber  i)rimus, 
de  menfuris  Romanis '  (p.  9)  ;  '  Liber  secundus  de  Mensuris  Cirajcis ' 
(P-  75)  j  'Liber  tertius,  de  Pondere  rerum  quas  metimur '  (p.  144); 
'Liber  quartus,  de  Ponderibus  Romanis'  (p.  188);  'Liber  quintus, 
de  Ponderibus  Grsecis'  (p.  219).  The  book  is  also  valuable  to  the 
student  of  Roman  and  (ireek  numerals,  and  of  the  various  symbols 
of    measures.     Such   works  explain    the   origin   of    certain   systems   of 


172 


RARA   ARITHMETICA 


iatt/attfft>ctt@c§6ffd/£tmer/ 

^nt>  $5funbcc5ett)t(?t/^tt  cl)** 

ten  mm  ^xb<ixn/W>^ifcn 


If  55 

tMelc^tot;  S,m<t. 
if56 

Fig.  86.   Title  page  of  the   1536  riese 


>" 


PRINTED    lUJCJKS  I73 

measures  employed   before   the  metric  system  was  developed,  and  of 
such  symbols  as  are  still  used  by  apothecaries. 

GEORGII 

AQRi  CO  LAB  MEDICI    LlBRl 

am({ucde  Menfuris  CT  Vondefibusjnqm 

bui  plxra(iue4  B  V  D  AE  O    CT 

P  OB.T  J  O  parum  amnMiucrf4 

diligftrttrexfummur* 

Oj^m  nunc  primum  in  kcm  ^iditunu 


p  A  R  I  S  I  I  S> 

nxcudeUl  ChrjjlwwJ  Wfck(HJ,wi  uicoUcOs 

bt/eoJubfcuto  B^ilecnft,    Anno 

W,D.  XXXIU. 

Fig.  87.    Title  page  of  the  first  edition  of  agricola 

0/her  works  0/1533.  Agrippa,  p.  167,  153 1  ;  Apiamis,  p.  62  ;  Jorda- 
nus,p.62,  1496;  Riese,p.  139,  1522  ;  Schonerus  (editor),  p.  178,  1 534.: 
Vincento,  p.  140,  1522;  Anonymous,  'Libretto  de  Abaco,'  Vemce ; 
Anonymous  (sometimes  attributed  to  Regiomontanus),  p.  178.  i534- 


174  RARA   ARITHMETICA 

GIOVANNI    MANENTI.     Ed.  pr.  1534.  Venice,  1534. 

ZuAN  Manenti.    a  Venetian  mathematician  of  the  sixteenth  century. 

Title.    See  Fig.  88. 

Colophon.  '  In  Vinegia  per  Giouan'  Antonio  di  Ni-//colini  da 
Sabio  A  Inftantia  de.  M.//  Zuan  Manenti.  Nelli  Anni  del  //  figno- 
re.  MDXXXIIII.//del  Mefe  di  Genaro.//  Neffuno  ardifca  Stam- 
par  quefte  Tariffe//  de  cambii  &  de  diuerfe  cofe  fotto  pe//na  de 
excommunicatione  late  //  fententie  come  nel  Priui-//legio  fe  con- 
tiene.//  M  D  XXXIIIL'    (Large  woodcut.)    (F.  400,  v.) 

Description.  12°,  7.5  X  13.9  cm.,  the  text  being  5.4  X  10.6 
cm.    402  ff.  (2  blank)  unnumb.,  21-26  11.    Venice,  1534. 

Editions.  There  was  no  other  edition. 

As  the  title  indicates,  this  is  a  set  of  tables  of  exchange,  and  it  was 
intended  for  the  use  of  Venetian  bankers  and  merchants. 

GIOVANNI    SFORTUNATI.     Ed.  pr.  1534.     Venice,  1534. 

Johannes  Infortunatus.   An  Italian  arithmetician,  bom  at  Siena  c.  1 500. 

Title.    See  Fig.  89. 

Colophon.  '  Stampata  in  Vinegia  per  Nicolo  di  Ariftotile  // 
detto  Zoppino.//M.D.  XXXIIIL'    (F.  129,  r.) 

Description.  4°,  15.4  X  21  cm.,  the  text  being  13.2  X  16.9  cm. 
129  ff.,  40-41  11.    Venice,  1534. 

Editions.  Venice,  1534,  4°  (here  described);  ib.,  1543,  4°; 
ib.,  1544  (colophon  1545,  see  p.  I77),  4°;  ib.,  1545,  4°  (p.  177); 
s.  1.,  Venice,  c.  1550;  ib.,  1561,  4°  (p.  177);  ib.,  1568.  The 
privilege  is  dated  1532. 

Sfortunati  wrote  his  treatise  along  the  lines  followed  by  Borghi  and 
Feliciano,  and  in  his  preface  he  acknowledges  his  indebtedness  to  them 
and  to  '  Maeftro  Luca  dal  Borgo  dell'  ordine  di  fanto  P'rancefco  '  (p.  54) 
and  to  the  'operetta  di  Fihppo  Caladri  Cittadino  Fiorentino  '  (p.  47). 
Like  these  authors,  he  was  a  popular  writer,  as  the  seven  editions  of 
his  book  go  to  prove.  His  work  is  fairly  complete  as  to  the  opera- 
tions with  integers  and  fractions,  and  is  satisfactory  as  to  the  examples 
illustrating  the  Italian  business  life  of  the  sixteenth  century.  The 
treatise  closes  with  some  work  in  practical  mensuration  and  some  mer- 
cantile tables. 


PRINTED    BOOKS 


175 


{^r  A  R  I  F  F  A^V 

DE  CAMBI  E  ALTRO 

Copofia  per  ZuaMcfncnti. 


Ccn  rriuilc^io  del  TlluH.Senatovcnero 
th'altrichcZ.Manoti  infra  anni  X  ftl 
par  nc  fcr  ftapar  no  la  polfj,  foTo  Ic  pc 
ne  cotenute in  quellcM  D  X  X XIIII, 


Fig.  88.    Title  page  of  manenti 


176 


RARA   ARITHMETICA 


;^    INVOVO  hVlAE^ 

,lBRO    Dl     ARITHMETICAL 

rntimbto:  NuouoLiimc  impoche  moltc  ^pofitfo 

ni  chcperaltn  autori:fono  falfajuete  coclufe;  in 

quefto  fi  emedano:&  caftiganorcon  chiare;  la 

dde:&  aperte  dimofl:rationi:molto  bene  di 

fcnfle  j&  ventillate.Co  vno  breac  trattato 

di  Geometria:per  quato  d  vnopratico 

Agrimenfore  fi  conucga,con  ta/ 

uole  da  r  cmporre  Ic  corde:da  mi 

furarcla  lenuta  di  ciafchuHd 

botte.&  eriam  liftaggfuox 

lidamifurareglifce 

mi  di  quelle* 

Compofto  per  lo  acutiffimo  prcfcrotato're.dene 

Archimedfanc&EacIidiane  dottrine 

GIOVANNI    SFORTVNATI 
DA  SIENA. 

CVM  GRATIA  ETPRIVILEGIO* 

M*    D.    XXXIIII- 


yyify  ^ 


:t 


Fig.  89.    Title  page  of  the  first  edition  of  sfortunati 


PRINTED    BOOKS  177 

GIOVANNI    SFORTUNATI. 

Ed.  pr.  1534.  Venice,  1544-45. 

See  p.  174. 

Title.  The  title  page  is  practically  identical  with  that  of  1534, 
except  for  the  date  :  M.  D.  XLIIII. 

Colophon.  '  Stampata  in  Vinegia  per  Bernardino  de  Bindoni 
//Milanefe  Anno  domini.  M.  D.  XLV.'    (F.  129,  r.) 

Description.  4°,  14.8  X  20  cm.,  the  te.xt  being  12.2  x  16.7  cm. 
2  ff.  unnumb.  +  127  numb.  =  129  ff.,  40  11.    Venice,  1544-45. 

Editions.  See  p.  174.  It  will  be  noticed  that  the  dates  in  the 
colophon  and  on  the  title  page  do  not  agree. 

See  p.  174. 

GIOVANNI    SFORTUNATI.     Ed.  pr.  1534.      Venice,  1545. 

See  p.  174. 

Title.  The  title  page  is  practically  identical  with  that  of  the 
1534  edition. 

Colophon.  Tn  Vinegia  per  Giouan' Antonio  &  Pietro  fratel-//li 
de  Nicolini  da  Sabio.  Ad  inftantia  di//Giacomo  da  Coneano 
libraro  a  fan//Fantin.    M:  D.  XLV.'    (F.  129,  r.) 

Description.  4°,  15.4  X  21.2  cm.,  the  te.xt  being  12.9  X  16.8 
cm.  The  text  is  practically  identical  with  that  of  the  1534 
edition. 

See  p.  174. 

GIOVANNI    SFORTUNATI.     Ed.  pr.  1534.      Venice,  1561. 

See  p.  174. 

Title.  The  title  page  is  practically  identical  with  that  of  the 
1534  edition. 

Colophon.  '  In  Venetia  per  Francefco  del  Leno,// M  D  LXI.' 
(F.  129,  r.) 

Description.  4°,  15.4  X  21.6  cm.,  the  text  being  12.9  X  17.3 
cm.    5  ff.  blank  +  129  numb.  =  134  ff.,  41-43  11.    Venice,  156 1. 

See  p.  174. 


178  RARA   ARITHMETICA 

ANONYMOUS.    (Schonerus  editor.) 

Ed.  pr.  1534.  Nurnberg,  1534. 

Johannes  Schonerus  (Schoner)  was  born  at  Karlstadt,  near  Wlirzburg, 
January  i6,  1477,  and  died  at  Nurnberg  January  16,  1547.  He  was  a 
preacher  at  Bamberg,  and  later  (i  526-1 546)  a  teacher  of  mathematics  in  the 
Aegidiengymnasium  at  Nurnberg,  in  which  Melanchthon  took  such  interest. 

Title.    See  Fig.  90. 

ColopJion.  '  Norimbergae  apud  lo.  Petreium,// Anno  M.  D. 
XXXIIIL'    (F.  32,  r.) 

Description.  4°,  15  X  20.9  cm.,  the  text  being  1 1.6  X  15.6  cm. 
32  ff.  unnumb.,  29-43  11.    Nurnberg,  1534. 

Editions.  This  is  merely  an  edition  of  the  anonymous  medi- 
eval 'Algorithmus  Demonstratus,'  with  notes  by  Schonerus.  As 
might  be  expected,  therefore,  it  is  purely  theoretical,  being  a 
late  variation  of  the  Boethian  works.  In  the  preface  Schonerus 
speaks  of  it  as  the  'Algorithmus  Demonftratus  incerti  autoris.' 
De  Morgan  thought  that  it  might  have  been  written  by  Regio- 
montanus,  but  he  was  wrong  in  asserting  that  Schonerus  attrib- 
uted it  unquestionably  to  him.  As  a  matter  of  fact  the  authorship 
goes  back  at  least  to  the  fourteenth  century.  There  is  said  to 
have  been  an  edition  pubhshed  at  Nurnberg  in  1533,  attributed 
to  Regiomontanus,  but  I  have  not  seen  it. 

JOHANN    ALBERT.     Ed.  pr.  1534.  Wittenberg,  156 1. 

A  Wittenberg  Rechenmeister  of  c.  1 500-1 565. 

Title.  '  Rechenbuchlin  //  Auff  der  Federn/  Gantz  //  leicht/ 
aus  rechtem  Grund/  In  //  Gantzen  vnd  Gebrochen/  Neben  //  an- 
gehefftem  vnlangft  ausgelaffnem  Buch-//lin/ Avff  den  Linien/ 
Dem  einfel-//tigen  gemeinem  Man/ vnd  anhe-//benden  der  Arith- 
metica//zu  gut.//Durch  Johann.  Albert/// Rechenmeifter  zu 
Wittembergk/zufamen//bracht.  Auffsnew/mit  allemvleis  vber- 
//fehen/  gemehrt  vnd  gebeffert/  //zum  dritten  mal.//Wittem- 
berg.//  I  561.'     (F.  i,  r.) 

Colophon.  '  Gedruckt  zu  //  Witteniberg/  //  durch  Geor-//gen 
Lawen//Erben.// 1561.'     (F.  ii8,  v.) 


PRINTED   BOOKS 


179 


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Fr;.  90.    'I'lTLE  PAGK  (JF  TiiL:    1534  Algorlthrnvs  Dcmonstrat7s 


i8o  RARA   ARITHMETICA 

Description.  8°,  9.4  X  15.1  cm.,  the  text  being  6.^  X  10.8  cm. 
120  ff.  unnumb.  (two  blank),  7-24  11.     Wittenberg,  1561. 

Editions.  Wittenberg,  1534;  ib.,  1541,  12°;  Frankfort,  1541 
(the  colophon  is  dated  1542),  8°;  Wittenberg,  1553,8°;  ib.,  1554; 
Frankfort,  1558,  8°;  Magdeburg,  1559,  8°;  Wittenberg,  1561, 
8°  (here  described);  ib.,  1564,  8°;  Magdeburg,  1579,  8°;  Witten- 
berg, 1586,  8°.  That  this  edition  was  revised  in  1541  appears 
from  the  dedication,  which  is  dated  'im  taufent/ f Unff-//hundert/ 
ein  vnd  vier-//tzigften  Jar.'  Murhard  also  mentions  an  '  Intro- 
ductio  Arithmetices,'  Cologne,  1542,  8°. 

Although  from  the  title  it  would  seem  that  algorism  ('  Auff  der 
Federn ')  is  emphasized,  counter  reckoning  ('  die  Species  auff  den 
Linien ')  is  first  described  (ff.  A  3-F  2).  This  is  followed  by  the  second 
part,  the  algorism  :  '  Das  Ander  Rechenbfichlein/  auff  der  //  Feder/ 
auffs  aller  kiirtzeft  //  vnd  leichteft  ver-//faffet '  (f.  F  3).  In  each  part 
there  are  many  commercial  problems,  and  the  book  ranks  as  one  of 
the  most  practical  of  its  day.  It  is  a  valuable  source  of  information  as 
to  the  commercial  activities  of  its  time. 

Other  works  of  1^34.  Borghi,  p.  16,  1484;  Bradwardin,  p.  61, 
1495;  Jordanus,  p.  62,  1496;  Kolross,  p.  164,  1530;  Ortega,  p.  93, 
1512;  Peurbach,  p.  53,  1492;  Reichelstain,  p.  169,  1532;  Rudolff, 
p.  151,  1526  ;  Wolphius,  p.  154,  1527  ;  Rabbi  Elias  Misrachi,  rrN'7D 
"iCDQn  or  "iCDan  "ICD  (perhaps  1533?  1532?),  Constantinople,  4° 
(another  edition,  with  a  Latin  translation  by  Schreckfuchs,  and  a  com- 
mentary entitled  "ICDDH  PrN'pD  *lV:4p  by  Mlinster,  Basel,  1546,  4°). 

GIOVANNI    MARIANI.     Ed.  pr.  1535.  Venice,  1580. 

ZuANE  Mariani.    a  Venetian  arithmetician  of  the  sixteenth  century. 

Title.  '  Tariffa  perpetva  //  Con  le  ragion  fatte  per  fcontro  di 
//  qualunque  Mercadante  fi  voglia,//  che  dimoftra  quanto  monta 
ogni  //quantita  de  cadauna  mercantia  ad  //ogni  pretio,  si  a  pefo 
come  a  nume//ro.  Buona  per  ogniuno  in  Venetia,//  Dalmatia,  & 
altri  luoghi;  nelli  quali  //  fi  ragiona,  &  fi  fpende  a  moneda//Vene- 
tiana.  Et  e  buona  per  Verona,// Breffa,  Bergamo,  Milan,  Cre- 
mona,// Mantoua,  &  altri  luoghi  done  fi  //  ragiona,  &  fi  fpende 
a  moneda  Im-//periale,  &  Breffana  :  Con  la  reduti6//di  moneda 
Venetiana    in    mone-//da    Imperiale,    &    della    imperiale    in  // 


PRINTED    BOOKS  kSi 

Vcnctiana.  Et  e  buona  a  ridurrc // ogni  forte  do  ori  in  moncda 
cor-//rente,  si  Vcnctiana  come  Im-//pcriale  :  &  Brcffana  ad  // 
ogni  precio.'     (¥.  i,  r.) 

Coloplioii.  'Stampata  in  Vcnctia  per  gli  Hcredi  di  //  l^Vanccfco 
Rampazetto.  //  Ad  inftantia  de  lAutore  Zuane  //  Mariani.// 
L'Anno  .M.  D.  LXXX.'    (F.  299,  r.) 

Description.  12°,  8.5  X  15.2  cm.,  printed  in  double  columns, 
each  being  2.6  X  13.3  cm.  279  ff.  numb.  +  20  unnumb.  =  299  ff., 
32  11.    Venice,  1580. 

Editions.  This  is  apparently  the  first  of  two  books  by  Mariani, 
the  various  editions  appearing  as  follows :  Venice,  1535;  ib.,1553; 
ib.,  1559;  it).,  1564,  8°;  ib.,  1567;  ib.,  1569,  16°;  ib.,  1572; 
ib.,  1575;  ib.,  1579;  ib.,  1580,  12°  (this  edition);  ib.,  1591,  16°. 
The  second  book,  also  a  Tariffa,  appeared  three  times  at  Venice, 
viz.  in  1538,  1555,  and  1558.  That  these  were  different  works 
I  know  only  from  such  bibliographers  as  Riccardi  and  Libri. 

Like  other  books  with  the  same  title,  this  is  simply  a  set  of  tables 
for  the  use  of  merchants.  It  includes  both  interest  and  exchange  tables, 
and  is  adapted  to  the  needs  of  Northern  Italy. 

Other  works  0/1535.  Agricola,  p.  171,  1533;  Angelus  Mutinens, 
p.  140,  1525  ;  Finaeus,  p.  160,  1530-32  ;  Grammateus,  p.  123,  1518; 
Kobel,  p.  102,  15  14  ;  Reisch,  p.  82,  1503;  Riese,  p.  141,  1522  ;  Ton- 
stall,  p.  134,  1522  ;  Torrentini,  p.  76,  1501  ;  Pedro  Melero,  'Compendio 
de  los  numeros  y  proporciones,'  Saragossa,  4°. 

HUDALRICH    REGIUS.     Ed.  pr.  1536.         Freiburg,  1550. 

A  German  teacher  of  the  first  half  of  the  sixteenth  century. 

Title.  '  Vtrivs-//qve  arithme-//tices  epitome,  ex  variis  // 
authoribus  concinnata,  per  //  Hvdalrichvm  //  Regium.//  Nvnc 
Tertio  omnia  //  diligenter  reuifa  8z  emendata.//  Friburgi  Brif- 
goiae,//  Stephanas  Grauius  excu-//clebat.  Anno  //  M.  D.  L.' 
(F.  I,  r.) 

Colophon.  '  Fribvrgi  Brisgoiae, // Stephanus  Grauius //excu- 
debat,//Anno    M.  D.  L.'    (F.  104,  v.) 

Description.  8°,  10  X  15.3  cm.,  the  text  being  7  X  ii.i  cm. 
I  f.  unnumb.  +  103  numb.  =  104  ff.,  17-22  11.     Freiburg,  1550. 


i82  RARA   ARITHMETICA 

Editions.  Strasburg,  1536,  8°;  Freiburg,  1543,8°;  ib.,  1550, 
8°  (here  described). 

This  work  was  intended  for  the  Latin  schools.  It  is  only  slightly 
practical,  and  as  compared  with  a  book  like  that  of  Gemma  Frisius  it 
is  reactionary.  The  first  part  (to  f.  48)  treats  only  of  Boethian  arith- 
metic, the  theory  of  numbers,  closing  with  the  words  :  '  Hactenus  de 
numerorum  Theo-//rijs,  nunc  de  eorundem  //  Praxi.'  The  practical 
part  gives  the  operations  in  the  usual  style  of  the  Latin  writers  of  the 
time,  and  closes  with  several  pages  on  the  use  of  counters. 

Other  tvorks  0/1536.  Boethius,  p.  27, 1488 ;  Bradwardin,  p.  61 ,  1495  ; 
Budaeus,  p.  99,  1514;  Feliciano,  p.  148,  1526;  Ortega,  p.  93,  1512; 
Peurbach,  p.  53,  1492  ;  Riese,  p.  139,  1522;  Torrentini,  p.  76,  1501 ; 
Christiernus  Torchillus  Morsianus,  '  Arithmetica  practica,'  Basel,  8°,  with 
subsequent  editions,  ib.,  1538,  1553,  8°  (but  see  p.  159,  1528)  ;  Georg 
Walckl,  '  Die  Walfch  practica/  gezoge  auf5  der  kunft  der  Proportion,' 
Strasburg  (Niirnberg  ?),  8°;  Rycharde  Benese,  'This  boke  sheweth  the 
maner  of  measurynge  of  all  maner  of  lande,  as  well  of  woodlande,  as 
of  lande  in  the  felde,  and  comptynge  the  true  nom^re  of  acres  of  the 
same,'  London,  4°.  (De  Morgan  includes  this  book  because  of  its  com- 
putations and  early  mathematical  tables.  Subsequent  editions  appeared 
in  1537,  1540,  c.  1558,  1562,  and  1564.)  L.  Culman,  '  Wie  iunge  und 
alte  Leut  recht  petten  sollen,'  Niirnberg,  8°;  ib.,  1537. 

ABRAHAM   BOSCHENSTEYN.    Ed.  pr   1536.    S.  1,1536. 

The  son  of  Johann  Boschensteyn  (see  p.  99). 

Title.  '  Ein  niitzlich  //  Rechenbilch-//lin  der  Zyffer/  darauC 
ein//yeder/  durch  fein  aygen  f^eyfz  mit//kleyner  hliff/ lernen 
mag  anfeng//klich  rechenen/  Aufzgange  durch  //  Abraham 
Bofchenfteyn/  Vnnd  //  yetzo  ziim  dritten  mal  mit  fleyfz  //  vber- 
fehen  vnnd  Corrigiert/  mit  //  erlichen  zugethanen  Exem//plen/ 
Durch  Johann  //  Bofchenfteyn/  den  alte.//  M.  D.  XXXVI.' 
(F.  I,  r.) 

Description.  8°,  9.5  x  14.5  cm.,  the  text  being  6.8  X  11.9  cm. 
40  ff.  unnumb.,  20-23  11.    S.  1.,  1536. 

Editions.  There  was  no  other  edition,  but  the  book  seems  to 
have  been  written  in  1530,  for  the  dedicatory  epistle  is  dated 
'  Gebe  am  19.// tag  Aprilis/ An-//no  h".  im  30 //jar  d'  min// 
derenn  //zal.'    The  work  is  very  rare. 


PRIiNTED    BOOKS  183 

In  the  epistle  the  author  mentions  his  father's  work  (p.  99)  :  '  Wie- 
wol  meyn  Herr  vatter/  herr  Johan  Bollhenfteyn  vor  i  7.  jaren  auch  tier 
gleych  zu  Aiigfpurg  inn  den  Truck  mitgetheylt  hat/  \  nnd  ziim  drittenn 
mal  getruckt  worden.'  Seventeen  years  before  1530  was  15 13,  when 
Johann's  book  was  probably  written,  since  it  was  published  in  15  14. 

Abraham's  work  is  not  much  of  an  improvement  on  his  father's, 
and  resembles  it  in  many  respects.  It  gives  seven  '  Species,'  including 
'  Duplicatio '  and  '  Mediatio,'  as  Johann's  work  had  done.  The  ])rin- 
cipal  additions  are  in  the  applied  problems. 

GIEL  VANDER  IIOECKE.    Ed.  pr.  1537.    Antwerp,  1537. 

A  Dutch  arithmetician  of  the  first  half  of  the  sixteenth  century. 

Title.    See  Fig.  91. 

Coloplioti.  '  Gheprent  Thantwcrpcn  op  die  Lombaerden  vcflc 
//  teghen  die  gulden  hant  ouer  by  mi  Symon  Cock.//  Int  Jaer 
ons  Heeren  ;  M.CCCCC.  ende  //  XXXVII.  den.  ix.  dach  Febrii- 
arij.'    (F.  180,  v.) 

Description.  8°,  9.4  X  14.7  cm.,  the  te.xt  being  "j .6  y^  12.8 
cm.  5  ff.  unnumb.  +  176  numb.  =  181  ff.,  26-32  11.  Blackletter, 
as  shown  in  Figs.  91,  92.    Antwerp,  1537. 

Editions.  So  far  as  I  know  this  edition  of  1537  is  the  first, 
the  date  15  14  in  the  British  Museum  catalogue  being  evidently 
an  error  for  1 544.  The  book  was  again  published  by  the  same 
printer  in  1 544,  and  there  was  an  edition  in  1 548.  (SQcBil'/iot/ieca 
Mathematica,  1906,  p.  211.) 

The  author  begins  with  the  fundamental  processes  with  integers,  con- 
sidering the  subject  in  a  practical  way.  He  then  considers  the  same 
processes  with  counters  on  the  line  abacus.  This  preliminary  work  is 
followed  by  chapters  on  denominate  numbers,  fractions,  the  nde  of  three, 
roots,  and  the  mediaeval  proportion.  The  second  part  of  the  treatise  is 
devoted  to  algebra  and  the  applications  of  arithmetic.  The  w'ork  closes 
with  a  brief  treatment  of  mensuration.  It  is  especially  noteworthy  on 
account  of  the  early  use  of  the  plus  and  minus  signs,  not  heretofore 
noticed  by  writers  on  the  history  of  the  subject.  There  is  no  other  Dutch 
book  of  this  period  that  makes  as  much  use  of  these  signs  (see  Fig.  92), 
and  Vander  Hoecke  should  be  recognized  as  among  the  pioneers  in  appre- 
ciating their  value  in  connection  with  algebraic  quantities.  (See  Gram- 
mateus,  p.  125,  1518,  who  used  them  in  a  similar  way.) 


i84  RARA   ARITHMETICA 


pttfeatt^ukn/ al0  4Dicnumeraae oani)m0^(tale 

iC^((  rc0uleoan  2);pm  mt0l)c^(cle  en  uit  gbthjuhu 
tOtdt^ulc  uan  cmoalfc^e  t>ofitte  Im  met  oalfcepo^ 
fttienoan  dmcerfrt^en  0^crotctite/matc/cR0^cli)c, 
C«^oc  Ijebiip  Die  etiel  rc0ul€  ^^O0/  t)ie  Ian0^  octto^t: 
0^01  ]^(ejf^  0t)ctt)(cf7/  todchc  re0ulc  10  die  Dojtcoati  al(c 
quefhat^efc  rc0ul(^d)&p  mct^aer  fp((im/a!0f^tus 
meratto/aDbitto/fubtractio/multlpKcatio/enDaiiao/ 
(It  met  ^cr  eoaiiacie  oft  qWmxghmfi  met  Die  xe^ue 
le  Der  quantitept/annejc  Der  re0U(en<ro0. 
Zteuanal(ecoopmaffappe/al0oanla6ett/rp^me/(n 
inercerie/i)temcoer(ooptt)t0cn){c^ee/mate/dtnomer/ 
en  Oatfelue  00c  te  merchf  2)oertteit0uleoanp;ta(tihe 
QE^ter(0u!(t)dn0^cfclfrap/mett)iiieerf(^enmie00e# 
<^oci}tefehieit0utcmetDiueetfceh(nle00e/metdmet^ 
€n^terc0u1eDanrmaelt)eelm0eofitom       (fcetide 
fieum0en  UitictranfpoonttnDanDenlaniutVjp^ofitel^ 
t^oeralleontfang^erooa  fubuencim, 
X)k  re0ule  oan  man0elm0e,  oan  affapertltatt  0oude/ 
en  oan  6luer/en  oanf  nannen  Dan  mapenen^ 
^)c  ^b;iHe  oandwi^m'oeDe  ofcotfierrotOc/rn  fitt  t){^ 
X)tp%Qtmtnm  (cnenf?tcHelant0teme<  (tetjaHie 
ten  alfo  tod  Oat  onbe0aTi0e!ijc  10  miOt0  Ocn  ty ater/oft 
dnticr0/al0  bat  be0an0bel^r  to. 
6!ietalcwleerteltDerfaemtmet0rootertta^Ptcliept/ 
bi<i»telt0oanb§boccHe.  «ftt0bep;tent<p»nttoerpctt 
op  bte  tomba6rbci)r/?e.^pmi^f  mon^Tcdu 

Fig.  91.  Title  page  of  vander  hoecke 


PRINIKI)    BOOKS 


C/tcmwtlt)i  aftrcchm  Ix  ly  uanl^  ^  foat) 
fcftrtfjcptjcticquntjiatmcccti  /^^  tflcrnamuiti 
plicccrt  Hjt  icn  ouaUiacc  \mm\  AuHcrc  ncocmt  i  hit 
trtttmu  -rl^^^fl  I'li  notcl^tJacrafio^i 

^'(rratioitalc. 
C^Oif  <^Taliotia!cicbt  tact— oft  mct  +  itacr  Orn 

cpfc!juant3(nU)a:l{c/uIfit«clU0c  j  o^"l^  \ 
refill:   i  _iv;^ 

14  1 

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CttcmailOiaftrccltciU^  ^  »anl^) —  y^"^^ 

CI  tern + »an  ^tft — ua» — fubtratjccrt  ciitjc  + 
uan — off  — .uan  -i-  aOiDcm  intjcfubtractic  fo  ucr* 
re  dt0  fi  atiDfcrlKftui  eft  fiibtraiicrrltr. 

C  rl^ultiplifaac  mOrii  iV:  oanraficitalcit. 
(jTriDilOimultipliffTcn  intiruikfccnjcft  tt\t£hp 
^-AtnnctfMlcnaUcOcnomincrouaii  miOti  iiatu 
realotx  tt  multiplicmtt  mrt  fTmpclcit  nomimrfcc 
mott  oln  Den  nommcr  muUiplucrcn  mtttt  nualir  f  vt 
Df  0 1%:  ,Hlo  ruiltJi  multipltfctcr  iv  o  met  4  fo  ftt  4  in 
tincn  iv.  nuiltipliccert4  inbacr  fducn  cotmt  l<c  10  ro 
inu'.tipliccatomctio  cofti44l?ieriut  rrcft  l\:foc( 
IX  foe  pcc!  10  tx  9  flhcmultiplif  cert  met  4/  want  In  9 
io^Dttniu!tipliccertmet4  eoemtii  alooojai. 
WUt)unu!tjplifcrHV*8mct5  fo  mwltipUcccrtfcu 

Fig.  92.    From  vander  hoecke's  Arithmetica 


i86  RARA   ARITHMETICA 

Other  -works  0/1337.  Apianus,  p.  155,  1527  ;  Baeda,  p.  159,  1529  ; 
Benese,  p.  182,  1536;  Kobel,  p.  no,  15 14;  Rudolff,  p.  152,  1526; 
Wolphius,  p.  154,  1527  J  Andres,  p.  122,1515;  B.  C.  Symphorien  Cham- 
pier,  '  Libri  VII,'  Basel,  8°  (one  chapter  '  De  Arithmetica  ') ;  Culman, 
p.  182,  1536  ;  Anonymous, '  An  Introduction  for  to  lerne  to  reckon  with 
the  Pen  and  with  the  Counters  after  the  true  cast  of  Arsmetyke,  or 
Awgrym,'  St.  Albans. 

NICOMACHUS.     Ed.  pr.  1538.  Paris,  1538. 

Born  at  Gerasa;  flourished  c.  100  a.d.  He  was  a  neo-Pythagorean  phi- 
losopher and  mathematician,  and  attempted,  unsuccessfully,  to  do  for  the 
Greek  theory  of  numbers  what  Euclid  had  done  for  geometry.  Two  works 
of  his  are  extant,  this  treatise  and  a  '  Harmonices  Manuale.' 

Title.    See  Fig.  93. 

Description.  4°,  157  X  23.1  cm.,  the  text  being  9.4  X  17.7 
cm.    77  pp.  numb.,  32  11.,  Greek.    Paris,  1538. 

Editions.  This  is  the  first  edition  of  the  arithmetic  of  Nicoma- 
chus  in  Greek.  A  second  edition  appeared,  '  Explicata  per  Joach. 
Camerarium,'  at  Augsburg  in  1 5  54  (p.  263).  For  the  commentary 
of  lamblichus,  see  p.  188. 

The  arithmetic  of  Nicomachus  is  the  most  celebrated  of  the  few 
Greek  treatises  upon  the  subject.  It  was  written  during  the  decline  of 
Greek  learning,  and  is  not  a  work  of  great  merit,  being  chiefly  a  com- 
pilation of  the  general  number  theory  of  the  Pythagoreans.  There  are 
several  commentaries  upon  the  'Introductio,'  that  of  lambhchus  (c.  325 
A.D.,  see  p.  188)  being  the  best  known  of  the  ancient  ones,  and  that  of 
Camerarius  (see  p.  262)  being  the  most  important  one  of  the  Renaissance. 

After  a  philosophical  introduction,  Nicomachus  classifies  numbers  as 
even  and  odd,  and  the  odd  as  prime  and  composite.  Perfect,  excessive, 
and  defective  numbers  are  also  considered,  and  the  elaborate  system  of 
ratios  which  later  characterized  the  work  of  Boethius  and  the  medieval 
writers  is  given.  Polygonal  and  solid  numbers  and  proportions  are 
treated  in  the  second  part,  a  ratio  being  loosely  defined  as  '  the  relation 
between  two  terms,'  and  proportion  as  '  the  composition  of  ratios.'  The 
work  differs  essentially  from  Euclid  in  its  presentation,  being  inductive 
instead  of  deductive  in  treatment.  It  is  also  a  matter  of  interest  that  the 
first  multiplication  table,  the  '  mensa  Pythagorica '  of  medireval  writers, 
to  be  found  in  any  treatise  appears  here,  although  Hilprecht  found  them 
on  the  Babylonian  cylinders  of  about  2000  b.c.  The  best  edition  of  the 
works  of  Nicomachus  is  that  of  Hoche  (Leipzig,  1866). 


PRINTED    BOOKS 


187 


NIKOMAXOY    TEPA: 

2  IN  or    a'pigmhtikhz'  bibai'a    Ar'o. 

NICOMACHI    GERA: 

SINI   ARITHMETI- 
cae  libri  duo. 


J^'^fJr 


trat^    ^^ 


/iCP 


P  A  U  1  S  1 1  S. 

In  ofFicina  ChriRiani  Wechcli. 
M.  D.  XXXVIIL 

Fig.  93.   Title  page  of  nicomachus 


i88  RARA   ARITHMETICA 

GILLES    HUGUETAN.     Ed.  pr.  1538.  Lyons,  1538. 

A  Lyons  arithmetician,  bom  c.  1500. 

Title.    See  Fig.  94. 

Colophon.  '  Icy  finiffent  les  tables  des  comptes  compofees  et 
calculees  //  par  Gilles  huguetan  :  Et  imprimees  cheux  //  ledict 
Gilles  et  Jacques  huguetan // freres.    Lan//i538.'    (P.  115.) 

Description.  F'ol.,  21.3  X  32.9  cm.,  the  text  being  18.  i  X  27.3 
cm.     25  pp.  unnumb.  +  90  numb.  =  115  pp.,  59-^4  l^-     Lyons, 

1538. 

Editions.    There  was  no  other  edition. 

Because  it  is  composed  largely  of  multiplication  and  division  tables, 
and  of  other  tables  of  use  to  stationers  and  merchants,  this  work  is  not 
often  included  among  the  arithmetics  of  the  century.  It  should  be  so 
classed,  however,  since  the  first  eleven  folios  are  devoted  to  the  expla- 
nation of  the  fundamental  operations  both  with  written  numbers  and 
with  counters.  The  illustrations  of  counter  reckoning  are  striking,  the 
'gectz'  (counters)  being  represented  full  size.  The  book  is  one  of  the 
earliest  Lyons  arithmetics  in  which  the  line  abacus  is  mentioned. 

lAMBLICHUS.     Ed.  pr.  1538.  Arnheim-Deventer,  1668. 

Bom  at  Chalcis,  in  Ccele-Syria,  c.  283  ;  died  at  Alexandria,  c.  330.  He  was 
a  neo-Platonic  philosopher  and  a  voluminous  writer.  P'our  of  his  works  are 
extant,  this  introduction  to  the  arithmetic  of  Nicomachus  being  one. 

Title.    See  Fig.  95. 

Description.  4°,  15  x  19.8  cm.,  printed  in  double  columns, 
Greek  on  the  left  and  Latin  on  the  right,  each  column  being 
5  X  14.8  cm.  12  pp.  unnumb.  +  181  numb.  =  193  pp.,  34  H- 
Bound  with  the  Camerarius  edition  of  Nicomachus  (p.  262, 
1554).    Arnheim-Deventer,  1668. 

Editions.  There  was  no  other  edition  in  the  sixteenth  century 
than  that  of  1538. 

See  Nicomachus,  p.  186.  This  commentary  by  lamblichus  forms 
the  fourth  part  of  his  treatise  on  the  Pythagorean  philosophy,  the  greater 
part  of  which  is  still  extant. 

Other  works  of  1538.  De  la  Torre,  p.  41,  1489  ;  Glareanus,  p.  191, 
1539;  Mariani,p.  181,  1535  ;  Morsianus,  p.  182,  1536  ;  Peurbach,  p.  53, 


PRlNlM';i)    HOOKS  iSg 

?^LESTABLESDEDI<«i 

VERS    COMPTES,   AVEC   LEVRS  CANONS, 
calculccs parGlLLES    HVGVETAN, natif dc Lyon, 
Par  Icfqucllcs  on  pourra facilcmcnt  trouucr  Ics  Comptcs  tous  faictz,  tant  dcs  achats 
que  ucntcs  dc  routes  marchandifes.foic  en  gros.oucn  detail,  alaMcfurc,  ouau  Poix, 
a  la  Charge,  ou  au  Nombre. 

Lcs Tables  aulFi  du  fin  Dor  Sc  Dargcnt,  pour  fcauoir,  fcclon  que  Ic  Marc  de  billon 
ticndra  dc  fin.ou  daloy,combien  il  uauldradepoixdc  fiiiOr.ou  Dargcnt  fin. 

Deux  Tables  feruantz  aux  Librafrcs.Et  unc  Table  dc  Dcfpcncc,  a  fcauoir  a  tant  pour 

lour.combienondefpcndlan  S^leMoys.&TaraylbnduMoyscombicnrcuicntpour 

an  8C  pour  chafcunIour,S(:a  taiit  pour  An  comb  ien  ondcfpend  leMoys,SC  chafcun 

lour. 

La  manierc  dc  AuaIucr,ouRcduyrc  par  icelles Tables  toutes  Monnoyca,cn  liurcs, 

{o\z,&C  denicrs. 

J  L  A  R  T  8«:fcicnce  dcNombrcr,Adioufter,Soufb-airc,MuIupIier, 
&:Partir,par  le  comptc  dcsGectz, 


On  Ics  ucnd  aLyon,  a  Icnfcigne  de  la  Sphacrc,chcux 
Gillc$,8C  laqucs  Huguetan,frcrcs, 

iy3S 

Ficj.  94.   Title  page  of  huguetan 


I90  RARA   ARITHMETICA 

1492  ;  Psellus,  p.  168,  1532  ;  Riese,  p.  138,  1522  ;  Roche,  p.  130,  1520; 
Rudolff,  p.  160,  1530  ;  Tonstall,  p.  135,  1522  ;  Tomas  Klos, '  Algoritmus  : 

JAMBLICHUS 

CHALCIDENSIS 

Ex  Coele- Syria 

I  N 

NICOMACHI     G  ERA  SEN  I 
Arithmeticam  introdudioncm, 

E  T       D  E 

F     A     T     O. 

O^ncprimum  editusy  in  Latimm  fermonem  converfm, 

notis  ferpetuU  iUuftratus 

SAMUELE      TENNULIO. 

Accedic 

joachimi     camerarii 

Explicatio  in  duos  Libros  Nicomachi, 

cum  Indice  rerum  &  verborum  locuplcudimo. 
A     R     N     H     E     M     I     yE, 


Proftantapud  juh.   FRIDERICUM     HAGIUM, 

Davenmz  typisdefcripfir  WiLHEtMUS  WiSR, 
do  Id  c  lx  vii£. 

Fig.  95.   Title  page  of  the  1668  iamblichus 

to  iesth  nauka  Liczby,  Polska  rzecza  wydana  :  Przez  Ksiedza  Tomasza 
Klosa,'  Cracow  (reprinted  at  Cracow  in  1889)  ;  Eysenhut,  'Ein  kiinstlich 
Rechenbuch,'  Augsburg,  8°. 


PRINTIH)    lUJOKS  191 

HENRICUS    LORITUS   GLAREANUS. 

Ed.  pr.  1539.  Paris,  1543. 

LoKiTi,  LoRKTi.  15orn  at  Mollis,  canton  of  (llarus,  Switzerland,  in  June, 
14SS;  died  at  PVeiburg,  Kieisgau,  May  2S.  i  $(y^.  He  was  professor  of  mathe- 
matics and  philosophy  at  Basel  (1515-1521),  and  professor  in  the  College 
de  France,  Paris  (1521-1524),  and  later  taught  at  Basel  and  Freiburg.  He 
wrote  on  arithmetic,  music,  and  geometry. 

T/t/c.    See  Fig.  96. 


Dc  Vi.  Arith 

METICAE   PRACTI- 

CAE   SPECIEBVS,HEN- 

RICI     GLAREANI 

Epitome. 


P  ARISIIS 

Ex  officma  lacohi  GaxcUhfuh  in- 

figni  \ntiidi<£,€  regione  gym- 

nafii  Cameracenfis. 

M  4  • 

Fig.  96,    Title  page  of  the  1543  gl.-^reanus 


192  RARA   ARITHMETICA 

Description.  8°,  10.6  X  17  cm.,  the  text  being  67  X  12.9  cm, 
2  ff.  uiinumb.  +  21  numb.  =  23  ff.,  29  11.    Paris,  1543. 

Editions.  Glareanus  is  sometimes  mentioned  as  the  author 
of  two  works  on  arithmetic,  the  '  Isagoge  Arithmetica '  and  the 
work  here  described.  Under  the  former  title  the  following  edi- 
tions are  mentioned  by  various  writers :  Freiburg,  1 539,  8°  (unless 
Tropfke  is  correct  in  saying  there  was  an  edition  of  1538);  Paris, 
1554,  8°;  Lyons,  1554,  8°.  Under  the  above  title  the  following 
editions  appeared:  Paris,  1543,  8°  (here  described.  The  dedica- 
tion is  dated  'Friburgi  Brisgoiae  .  .  .  M.  D.  XXXVIII,'  so  there 
may  have  been  an  edition  as  early  as  1538);  Freiburg,  1543,  8"; 
Cracow,  1549;  Freiburg,  1550,  8°  (below);  Paris,  iS5i,8°(p. 
193);  Freiburg,  1555;  ib.,  1558,  8°;  Paris,  1558,8°.  The  biog- 
rapher of  Glareanus,  Schreiber  (Freiburg,  1837),  gives  these  as 
the  same  work. 

Besides  this  book,  Glareanus  also  published  an  '  Arithmetica 
et  musica  operum  Boethii  demonstrationibus  et  figuris  auctior,' 
Basel,  1546,  fol.,  and  a  '  Commentarius  in  Arithmeticam  et 
Musicam  Boethii,'  Basel,  1546,  fol.;  ib.,  1570,  4°. 

A  handbook  for  the  Latin  schools.  In  it  Glareanus  first  treats  of 
notation,  including  the  Greek,  Roman,  and  Arabic  systems;  then  of 
the  elementary  operations  with  integers ;  then,  briefly,  of  progressions 
and  proportion.    There  is  nothing  in  the  little  book  to  commend  it. 

HENRICUS   LORITUS   GLAREANUS. 

Ed.  pr.  1539.  Freiburg,  1550. 

See  p.  191. 

Title.  '  De.  VI .  Ari//thmeticae // Practicae Speciebvs // Henrici 
Glareani//P.  L.  Epitome.// Fribvrgi  Brisgoiae.// Cum  gratia  ac 
Priuilegio  Regio,//ad  annos  fex.'     (P.) 

Colophon.  'Apud  Friburgum  Brifgoicum // Anno  M.  D.  L.// 
Stephanus  Grauius //excudebat.'     (P.  T^.) 

Description.  8°,  10  X  15.8  cm.,  the  text  being  6.4  X  11.7  cm. 
2  pp.  unnumb.  +  2  blank  +  74  numb.  =  78  pp.,  22  11.  Freiburg, 
1550. 


PRINTED    BOOKS  193 

HENRICUS    LORITUS   GLAREANUS. 

Ed.  pr.   1539.  Paris,  i  55  i. 

See  p.  191. 

Title.  'Dc  fex  Arith-//mcticae  Practi-//cae  Speciebvs,//Hen- 
rici  Glarcani // Epitome.// Parisiis,// Apud  Guliclmum  Caucl- 
lat,  ill  pingui  Gallina,//c  rcgione  collcij  Camcraccnfis.//  155  i.' 
(F.   i.r.) 

Colophon.  '  Excudebat  Lutetiie  Parifiorum  Benedictu.s //Pre- 
uotius  Typographus  in  vico  Fre-//mentcll(),  fub  Stella  Aurea:// 
III.  Non.  lanuarij,// 1551.'     {F.  23,  r.) 

Description.  8°,  10. 4  X  16.  i  cm.,  the  text  being  6.4  x  11.9 
cm.     2  ff.  unnumb.  +  21  numb.  +  i  blank  =  24  ff.,  29  11.    Pari.s, 

1551- 

See  p.  192. 

HIERONYMUS   CARDANUS. 

Ed.  pr.  1539.  Milan,  1539. 

Geron'Imo  or  Girolamo  Cardano,  Jerome  Cardan.  Born  at  Pavia, 
September  24,  1501  ;  died  at  Rome,  September  21,  1576.  He  was  a  physi- 
cian and  professor  of  mathematics  at  Milan  (i 534-1 559)  and  professor  of 
medicine  at  Pavia  and  (i 562-1 570)  Bologna.  Later  he  was  a  papal  pen- 
sioner at  Rome.  He  was  one  of  the  most  acute  mathematicians  of  his 
century,  and  wrote  numerous  treatises  on  mathematics  and  natural  science. 

Title.    See  Fig.  97. 

Colophon.  'Anno  a  Virgineo  partu.// M.  D.  XXXIX. // 
lo.  Antonins  Caftellioneus  Me//diolani  Imprimebat  ImV/penfis 
Bernardini  //  Calufci.'    (Printer's  mark,  with  '  B.C.')  (F.  304,  v.) 

Description.  8°,  9.9  X  15.3  cm.,  the  text  being  8.2  x  12.7  cm. 
304  ff.  unnumb.,  33  11.    Milan,  1539. 

Editions.  Milan,  1539,  8°  (here  described);  Nurnberg,  1541; 
ib.,  1542.  See  also  p.  338,  1570.  A  1537  edition  is  given  by 
Villicus,  and  I  have  seen  it  mentioned  in  a  dealer's  catalogue ; 
but  I  think  the  date  a  misprint,  or  that  some  one  has  taken  the 
date  of  the  preface  instead  of  looking  at  the  colophon. 

This  is  one  of  the  most  pretentious  arithmetics  of  the  sixteenth  cen- 
tury, and  it  did  much  to  influence  the  advanced  teaching  of  the  subject. 
It  is  in  no  sense  a  practical  book,  having  been  written  by  a  mathematician 


94 


KARA   ARITHMETICA 


for  the  use  of  scholars.  It  opens  with  a  discussion  of  the  kinds  of  num- 
bers considered  in  arithmetic,  such  as  integers,  fractions,  surds,  and 
denominate  numbers.    This  is  followed  by  the  fundamental  operations 

HI  E  R  ON IMI 

C.CARDANI  MEDICI  MEDIOLA 
NENSrS,PRACTICA  ARITH> 

«necice,5C  Menfurandi  fingularis.Inqaa 

q\iz  prctcr  a!iasc6tinentur,vctfa 

pagina  Oemondcabit* 


Fig.  97.  Title  page  of  cardan 

with  these  numbers  and  a  treatment  of  proportion.  The  properties  of 
numbers  occupies  a  considerable  space  and  includes  much  of  the  ancient 
theory.  The  work  then  runs  into  algebra,  combining  this  with  arithmetic. 


PRINTED    IKJOKS  195 

'rhcre  arc  numerous  business  applications  in  the  treatise,  such  as  partner- 
ship, exchange,  profit  and  loss,  anil  mensuration,  but  these  are  treated 
from  the  theoretical  standpoint  rather  than  from  that  of  the  practical 
needs  of  the  merchant  class.  The  great  prominence  of  the  author  and 
the  scholarly  nature  of  the  work  account  for  the  various  editions  of  the 
book.    His  well-known  '  Ars  Magna'  (algebra)  appeared  in  1545. 


JOHANN    NOVIOMAGUS.     Ed.  pr.  1539  Paris,  1539. 

Neomagus,  Jan  Bronckhorst.  Horn  at  Nimwegen  in  1494;  died  at 
Cologne  in  1570.  He  was  for  a  time  professor  of  mathematics  at  Rostoctc. 
He  not  only  wrote  on  numbers,  but  edited  works  of  Bxda  and  Ptolemy. 

Title.    See  Fii;.  98. 

Description.  8°,  9.7  X  15.4  cm.,  the  text  bein<;  6."]  x  12.3  cm. 
117  pp.  numb.  +  2  unntmib.  =  119  pp.,  26-28  11.     Paris,  1539. 

Editions.  Paris,  1539,  8°  (here  described);  Cologne,  1544, 
8°  (below) ;  Deventer,  1551  (p.  197). 

The  book  was  intended  for  the  classical  schools.  It  sets  forth  the 
Roman  and  Greek  notations,  the  fundamental  operations  both  with  the 
Hindu  numerals  and  upon  the  line  abacus,  the  finger  notation  as  found 
in  the  works  of  Bteda,  the  astrological  numerals  of  the  Middle  Ages,  and 
the  Boethian  theory  of  numbers. 

Other  xvorks  of  i^jg.  Capella,  p.  66,  1499  ;  Peurbach,  p.  53,  1492  ; 
Ringelbergius,  p.  166,  1531  ;  Vogelin  (see  Peurbach,  p.  53,  1492);  Wol- 
phius,  p.  154,  1527  ;  Anonymous, '  Abacho  novo  con  il  quale  ogni  persona 
puol  imparar  Abacho  senza  che  alcuno  li  insegni,'  Venice  ;  Anonymous, 
'  An  introduction  to  algorisme,  to  learne  to  reckon  with  the  penne,'  Lon- 
don, 8°,  with  another  edition  ib.,  1581,  8°;  Juan  Gutierrez  de  Gualda, 
p.  167,  1531- 

JOHANN    NOVIOMAGUS.     Ed.  pr.  1539.      Cologne,  1544- 

See  above. 

Title.  '  De  Nvme//ris  Libri  II.  Qvo-//rimi  prior  Logifticen, 
&  ueterum  nu//merandi  confuetudinem :  pofterior//Theoremata 
numerorum  complecti-//tur,  aiitore  loan.  Nouiomago.// Nunc 
recens  ab  ipfo  autore  recogniti.// (Woodcut  with  motto:  'Discite 
Ivsticiam  moniti.')  Coloniae  loan.  Gymnicus  excudebat,// Anno 
M.D.XLIIII.'    (F.  I,  r.) 


196  RARA   ARITHMETICA 


DeNumcrisIi- 

BRI    DVO,  CLVORVM    PRIOR 

Logifticen  &  vecerum  numerandi  confuecudi- 

nem ,  polkrior  Theoremata  numerorum  com- 

pleditur,ad  dodifsimum'vfrum  Aadrc- 

am  Eggerdem  profefTorem 

Roftochienfem. 

Nunc  recens  in  lucem  emifsi  autliore 

loanne  Nouiomago. 


PARISIIS 

EroflRcfna  Chriftianf  Wecheli/ub  fcuts 

Bafilicnfijin  vico  Iacobaeo:&  lub 

PegafOjin  vico  Belloiiacendi 

M.  D.  XXXIX. 

Fig.  98.    Title  page  oe  noviomagus 


PRINTED    IK)C)KS  197 

Description.    8°,  10  X  14.8  cm.,  ihc  text  bcin|^-  G.j  X  1  1.5  cm. 
59  ff.,  25  11.     Cologne,  1544. 
Editions.    See  p.  195. 

This  is  merely  a  reprint  of  the  first  edition  (p.  195,  1539). 

JOHANN    NOVIOMAGUS.     Ed.  pr.  1539.     Deventcr,  1 55  i. 

See  p.  195. 

Title.  '  De  nvme//ns  libri  II.  qvo-/Avm  prior  logisticen,  ct 
//  ueterum  numerandi  confuetudinem  :  pofte-//nor  Theoremata 
numerorum  com//plectitur,  Autore  loan.//  Nouiomago.//  Nvnc 
recens  ab  ipso //autore  recogniti.// (Woodcut,  and  '  T.  B.  Pons 
lovis.')  Daventriae,// Theodoricus  Bornius  excudebat.// Anno 
M.  D.  LI.'     (F.  I,  r.) 

Description.  8°,  9.8  x  14.9  cm.,  the  text  being  6.8  X  10. 7  cm. 
50  ff.  unnumb.,  25  11.     Deventcr,  1551. 

Editions.    See  p.  195. 

Like  the  1544  edition,  this  is  a  reprint  of  that  of  1539. 

JODOCUS   WILLICHIUS.     Ed.  pr.  1540.    Strasburg,  1540. 

WiLKE,  WiixKE,  Wn,D.  Bom  at  Resel,  East  Prussia;  died  at  Lebus, 
November  12,  1552.  He  was  professor  of  Greek  (1540),  and  then  of  medicine, 
in  the  university  of  Frankfort  an  der  Oder. 

Title.    See  Fig.  99. 

Colophon.  'Argentorati  ex  officina // Cratonis  Mylii,// mense 
Sept.// anno //M.  D.  XL.'     (P.  125.) 

Description.  8°,  10.2  X  i  5.4  cm.,  the  text  being  8.2  X  i  i  .4  cm. 
2  pp.  unnumb.  +  123  numb.  =  125  pp.,  26  11.    Strasburg,  1540. 

Editions.    Strasburg,  1540,  8°  (here  described)  ;  ib.,  1545. 

This  is  a  book  intended  for  the  classical  schools.  It  is  written  chiefly 
in  Latin,  but  contains  numerous  extracts  from  the  (ireek.  It  is  based 
upon  Greek  models,  and  contains  several  quotations  from  Nicomachus. 
The  plan  of  treatment  is  catechetical  (see  Fig.  100),  and  it  is  interesting 
to  note  that  this  work  appeared  in  the  same  year  in  which  Recorde  may 
have  published  his  '  Ground  of  Artes '  in  England  (see  p.  213),  a  book 
in  which  the  author  also  adopted  the  catechism  form.  It  is  manifestly 
inspired  by  Roethius,  and  is  hair-splitting  in  theory  and  useless  in  prac- 
tice.   Willichius  begins  in  a  grandilocpient  style,  '  I)e  Arithmeticre,  quae 


198  RARA   ARITHMETICA 

Mathefeos  mater  eft,  finitione.'  The  history  of  arithmetic  has  few  more 
curious  examples  than  the  first  chapter  of  this  work,  with  its  learned 
references  to  Pythagoras,  Augustine,  the  Platonists,  and  '  an  Arab  phi- 
losopher named  Algebras.'  ('  Eadem  Autem  hodie  ab  authore  quodam 
x^rabe  Philofopho,  cui  nomen  erat  Algebras,  nomine  regularum  Algebrae 

lODOCI 

VVILLICHII 

RESELLIANI, 

Atitbmcticx  libri 


ARGENTORATI 
M,  D.  XL. 

Cm  Gr<iti4  V  vmilcgio, 

Fig.  99.    Title  page  of  willichius 

explicatur.'  P.  19.)  Willichius  follows  the  ancient  Greek  plan  of  divid- 
ing arithmetic  into  two  parts,  the  first  being  the  practical,  the  logistica 
of  the  classical  civilization  ('  vna  wpaKTiKr],  qua  fupputatio  dorneftica 
fit  .  .  .  apud  ueteres  a  ratiocinando  Aoyt^tK^  dicta  eft,'  p.  19),  and  the 
second  being  theoretical,  the  ancient  arithmetica.  ('  Eft  OewprjTLKr),  qua 
velut   fanctiora  myfteria   continentur,  &   haec    fola   intelligentia   animi 


PRINTED    BOOKS  199 

conftat,'  p.  20).  In  his  number  mysticism  lie  calls  unity  Jujiitcr  ('X'nitas 
ei't  lupiter,'  p.  22),  saying  that  others  call  it  Cupid,  others  Amicitia,  and 
others  Concordia,  and  (piotes  '  Zarathas  the  teacher  of  l'ythag(jras'  as 
calling  one  the  father  and  two  the  mother  of  numbers.  ('  Proinde  apte 
Zarathas  Pythagorae  prseceptor  dixit, /u-oi/uSueffe  numerorum  jjatrem,  SvuBa 

SO  ARITHMETICAE 

N  I  C  O  L  A  V  S. 

OYiieftalternumemparf  I  VST.  Ejt 
pctriter  imp<fr  ^uelk  paribus  iwp4r,Gr<fCi« 

f  «r .  g/ wro?.  Ejl  uutcitij  cum  primum  diuiditur:,mox  jit 

iniiuifibilis  ,ut  14.  » 8 .  zz.  N I C  O  L. 
Quomodo  cum  jinithorum  numerorum  ex(]uift* 
mmgifkrEuclideii  I  VST.  Sic:«^rt(CKiS 

rus  diuidens  par  ejl,fed  diuiforius  mox  impar  exwr 

get,  N I C  O  L.  Cur  id  nominis  iUi  inditum  eft  ? 

I V  S  T .  \deo,quod quilibet  eiiw  ordinis  numeri 

Cur  uoeatUY    pare$,fiHtfunt  perimparm  muUipUcationem:  ut 

pariter impar  yi^  ^er,  femrium^  bis  quin(^  denarium  conjiciunt, 

Verumfi  cui  altiui  contemplari  lAety  eundem  uo^ 

cabit  imparem  infua  qnmtitate  ^fed  parent  in  deno 

minatione.EJbexempligratiaJenarius,  cuiut  ds 

tera  pari  eft  quinariui,qui  quantitate,hoc  eft,  moif 

nadum  congregatione  eft  imparled  quia  i  binario 

denominaturypar  iudicabitur,  QU£  ratio  nomim 

ex  Boethio  colligitur:  Alia  autem  Euclidi  cffe  uide 

tur,  N  1  COL. SuntnehuicdeiUoaliquot theo 

SymboU  ex     remata  i  I V  S  T.  Quid ni  i  Wmm  ejl ,  Si mta 

parlterimpaei  menu  dimidium  impar  habuerit,  par  iter  impdT 

rL  eft  mtm  .  N<fltt  hie  dimkxat  cmemum ,  quoi 

mxh 

Fic  100.    From  the  arithmetic  of  wh.lichius 

autem  matrem,'  p.  22.)  The  contrast  between  this  work  and  that  of 
Gemma,  which  appeared  in  the  same  year,  is  very  marked.  It  must  be 
admitted,  however,  that  the  book  has  some  value  in  interpreting  the 
ideas  of  such  followers  of  Nicomachus  as  Boethius,  Jordanus,  and 
Faber  Stapulensis,  as  is  shown  in  Fig.  100.  It  has  at  least  the  merit  of 
having  been  written  for  beginners. 


200  RARA   ARITHMETICA 

GEMMA   FRISIUS.     Ed.  pr.  1540.  Wittenberg,  1542. 

More  properly  Gemma  Rainer  or  Regnier,  the  Frisian.  Born  at  Dockum, 
in  East  Friesland  December  8,  1508;  died  at  Louvain,  May  25,  1555.  He 
received  the  degree  of  Doctor  of  Medicine  in  I54i,when  he  abandoned  his 
mathematical  studies.  He  wrote  upon  astronomy  and  arithmetic,  and  his 
son  Cornells  was  a  contributor  to  the  former  science. 

Title.    See  Fig.  loi. 

Colophon.  '  Impreffum  Vitebergse  apud // Georgium  Rhau.// 
Anno  M.D.XLII.'    (F.  80,  r.) 

Description.  8°,  10  X  15.3  cm.,  the  text  being  7.8  X  15.3  cm. 
80  ff.  unnumb.,  23  11.    Wittenberg,  1542. 

Editions.  Antwerp,  1540,  8°;  Wittenberg,  1542,  8°  (here 
described);  ib.,  1543,  12°;  Paris,  1543  {.?);  Wittenberg,  1544, 
8°  (p.  202);  Paris,  1545  (Peletarius  edition);  Antwerp,  1547; 
Wittenberg,  1548;  Paris,  1549,  8°  (p.  202);  Paris,  1550,  8° 
(p.  203);  Wittenberg,  1550,  12°;  Antwerp,  1550;  Wittenberg, 
I55i>  8°  (p.  203);  Paris,  1551;  Antwerp,  1552,  8°  (p.  203); 
Paris,  1553,  8°  (p.  204);  Wittenberg,  1553;  ib.,  1555;  Lug- 
duni,  1556,  8°;  Paris,  1557;  Leipzig,  1558,  8°  (p.  204);  s.  a., 
but  c.  1558;  Paris,  1559;  Leipzig,  1559;  Paris,  1561,8°;  Wit- 
tenberg, 1561,  8°  (p.  204);  Leipzig,  1562;  Antwerp,  1562; 
Paris,  1562;  ib.,  1563  (p.  205);  Wittenberg,  1563,  8°  (p.  205); 
Cologne,  1565;  Leipzig,  1565,  8°;  Lugduni,  1566,  8°;  Witten- 
berg, 1567,  4°;  Paris,  1567;  Venice,  1567,  4°  (p.  205) ;  Leipzig, 
1568;  Paris,  1569,  8°;  Wittenberg,  1570,  12°;  Cologne,  1571, 
8°  (p.  206) ;  Paris,  1572  ;  Leipzig,  1572  ;  ib.,  1575,  8°  (p.  206)  ; 
Cologne,  1576,  8°;  Paris,  1578,  8°  (p.  207);  Wittenberg,  1579; 
Leipzig,  1580;  Antwerp,  1581,  8°  (p.  207);  ib.,  1582,  8°  (the  first 
with  Forcadel's  notes.?);  Wittenberg,  1583,  8°  (p.  208);  Paris, 
1585,  8°;  s.  1.,  1588;  Leipzig,  1588,  8°  (p.  208)  ;  ib.,  1591;  ib., 
1592,  8°  (p.  208);  Cologne,  1592,  8°;  Wittenberg,  1593;  Frank- 
fort, 1597  (the  only  German  translation.?).  There  were  numerous 
editions  after  1600.  Treutlein  {Abkandhmgen,  I,  18),  following 
Murhard,  says  there  Avere  at  least  twenty-five  editions  in  the 
sixteenth  century.  In  reality  there  were  more  than  twice  as 
many  ;  the  above  list,  probably  incomplete,  mentions  fifty-nine. 


PRINTED    HOOKS 


20I 


The  editions  of  Gemma  varied  i)ut  little  until  Pe 
several  years  before  the  former's  death,  added  his  notes, 
amplified  the  text,  but  they  made  no  changes  of  import: 

'J'his  was  the  most  jiopular  arithmetic  of  the  sixteenth  ce 
least  among  those  intended  ft)r  the  Latin  schools.    It  combined 


letaruis, 
These 
uice. 

ntury,  at 
the  older 


^/ 


ARITHME 

TICAE  PRACTI^ 

CAB  METHODVS  FACILIS, 
per  Gemma m.Frifium  MediV 
cum  ac  Mathematicutfi* 


VITEBERGAE.  M,   D.XLIf, 

Fig.  ioi.    TriLR  page  of  the   1542   g.kmma  erisius 


202  RARA   ARITHMETICA 

science  of  numbers  with  the  commercial  arithmetic  of  the  Italian  writers 
in  such  way  as  to  appeal  in  a  remarkable  degree  to  the  teachers  of  the 
period.  The  book  opens  with  a  discussion  of  the  various  fundamental 
operations,  presented  without  much  explanation  and  with  numbers  of 
relatively  small  size.  These  operations  include  the  subjects  of  dupla- 
tion  and  mediation  as  was  customary  in  the  Latin  books  of  that  time. 
The  author  closes  the  first  part  of  his  work  with  a  treatment  of  progres- 
sions and  the  rule  of  three.  The  second  part  is  devoted  to  fractions, 
the  sequence  being  the  same  as  with  integers.  The  third  part  includes 
such  common  rules  of  business  as  partnership,  alligation,  and  rule  of 
false,  together  with  roots  and  a  little  algebra.  The  fourth  part  treats 
of  proportion,  and  has  a  few  pages  on  arithmetical  recreations. 

GEMMA   FRISIUS.     Ed.  pr.  1540.  Wittenberg,  1544. 

See  p.  200. 

Title.  This  edition  is  substantially  the  same  as  that  of  i  542 
with  the  following  exceptions:  F.  i,  r.,  title  page:  '  Vitebergae 
Anno  M,D,XLIIII.' 

Colophon.    '  Anno  M,D,XLIIII.'    (F.  88,  r.) 

Description.  8°,  9.5  X  15  cm.,  the  text  being  7.9  x  11.3  cm. 
88  ff.  imnimib.,  23  11.    Wittenberg,  1544. 

See  p.  201. 
GEMMA   FRISIUS.     Ed.  pr.  1540.  Paris,  1549. 

See  p.  200. 

Title.  '  Arithmeticas  practicae  //  Methodvs  facilis  per  //  Gem- 
mam  Frisivm,  Medicvm  //ac  Mathematicum,  lam  recens  ab  ipfo 
//authore  emendata  &  multis  in  lo-//cis  infigniter  aucta.// Hvc 
accesservnt  lacobi  Pe-//Ietarii  Cenomani  annotationes  :  Eiufdem 
item  de  Fractio-//nibus  Aftronomicis  compendium  :  Et  de  cogno- 
fcedis  per  //  memoriam  Calendis,  Idib.  Nonis,  Feftis  mobilibus, 
&  loco  //  Solis  &  Lunae  in  zodiaco.//  (Woodcut  with  motto  :  '  In 
pingvi.  gallina.')  Parisiis.// Apud  Gulielmtim  Cauellat,  in  pingtii 
gallina,  ex // aduerfo  collegij  Cameracenfis.// 1549.'    (F.   i,  r.) 

Description.  8°,  10. 5  X  16.6  cm.,  the  text  being  7  x  12.7  cm. 
I  f.  unnumb.  +95  numb.  =96  ff.,  26  11.     Paris,  1549. 


PRINTED    BOOKS  203 

Editions.  This  is  the  ninth  edition  (see  p.  200),  and  the  first 
which  I  have  seen  with  the  notes  of  Peletarius,  although  these 
notes  bear  the  date  1545  (f.  yy,  r.)  They  are  here  given  in  an 
appendix  (Tacobvs  Peletarius  Lectori,'  f.  'j'j,  r.),  but  later,  as  in 
the  I  571  edition  (see  p.  206),  they  are  introduced  in  the  body 
of  the  text. 

See  p.  201. 

GEMMA    FRISIUS.     Ed.  pr.  1540.  Paris,  1550. 

See  p.  200. 

Title.  Except  for  the  date  (1550)  the  title  page  is  the  same 
as  that  of  the  1549  edition  (p.  202). 

Dcscriptio7i.  8°,  10.3  X  16.  i  cm.,  the  text  being  7.3  x  13  cm. 
I  f.  unnumb.  +  95  numb.  =96  ff.,  26  11.    Paris,  1550. 

See  p.  201. 

GEMMA   FRISIUS.     Ed.  pr.  1540.  Wittenberg,  1551. 

See  p.  200. 

Title.  '  Arithme//ticae  practicae  //  methodvs  facilis,  per  // 
Gemmam  Frifium  Medi-//cum  ac  Mathe-//maticum.//  (Wood- 
cut showing  computing  with  counters.)  Vvitebergae.//  Anno 
M.  D.  LI.'    (F.  I,  r.) 

Description.  8°,  9  x  14.2  cm.,  the  text  being  6.4  x  i  r.2  cm. 
87  ff.  unnumb.  +  i  blank  =  88  ff.,  23  11.    Wittenberg,  1551. 

See  p,  201. 

GEMMA    FRISIUS.     Ed.  pr.  1540.  Antwerp,  1552. 

See  p.  200. 

Title.  '  Arithme-//ticae  practicae  metho-//dus  facilis,  per  Gem- 
mam Frifium  Me-//dicum  ac  Mathematicum,iam  re-//cens  ab  ipfo 
auctore  emen-//data,  &  multis  in  locis  // infigniter  aucta.//Cor. 
Graphevs.//Si  numerandi  artem,  cunctis  ex  artibus  illam //Vel 
primam,  exacts  difcere  lector  amas,//  Hanc  gemma,  ingenio  fum- 
mus  quam  Gemma  libello  //  Hoc  paruo  includit,  carpito,  doctus 
eris.//  Sunt  conati  alij  prolixis  traderc  chartis  //  Hanc  artem,  at 


204  RARA   ARITHMETICA 

multis  non  placet  ille  labor.// Porro  hie  Gemma  fuam  gemmam 
fic  temperat,  ipfa // Vt  placeat  cunctis  commoditate  breui.// 
CAntuerpiae,  apud  Gregorium  Bontium  //  a  Csei.  Maieft.  libra- 
rium  admiffum.//Cum  gratia  &  priuilegio.//  15 52.'    (F.  i,  r.) 

Description.  8°,  9.5  X  13.9  cm.,  the  text  being  7.4  X  1 1.8  cm. 
4  ff.  unnumb.  +  ^6  mimb.  =  80  ff.,  28  11.    Antwerp,  1552. 

See  p.  201. 

GEMMA    FRISIUS.     Ed.  pr.  1540.  Paris,  1553. 

See  p.  200. 

Title.  Except  for  the  date  (1553)  the  title  page  is  the  same 
as  that  of  the  1549  edition  (p.  202). 

Description.  8°,  9.8  X  14.8  cm.,  the  text  being  6.8  X  12.7  cm. 
I  f.  unnumb.  +  95  numb.  =96  ff.,  26  11.     Paris,  1553. 

See  p.  201. 

GEMMA    FRISIUS.     Ed.  pr.  1540.  S.  1.  (Leipzig),  1558. 

See  p.  200. 

Title.  '  Arithme-//tica2  practicae  //  method vs  facilis,  per  // 
Gemmam  Frifium  Medicum  ac  //  Mathematicum  //  (Woodcut  as 
in  the  1542  edition.)    Anno  M.  D.  LVIII.'    (F.  i,r.) 

Description.  8°,  10.  i  x  15.4  cm.,  the  text  being  7.8  X  1 1  cm. 
87  ff.  unnumb.,  22-23  11-    S.  1.  (Leipzig),  1558. 

Editions.    See  p.  200. 

See  p.  201.  A  set  of  manuscript  notes  in  the  back,  in  a  sixteenth- 
century  hand,  gives  an  interesting  synopsis  of  arithmetic  as  taught  in 
the  universities  of  that  time. 

GEMMA    FRISIUS.     Ed.  pr.  1540.  Wittenberg,  156 1. 

See  p.  200. 

Title.  '  Arithme-//ticae  practicae  me-//thodvs  facilis,  per  Gem- 
//mam  Frisivm  Medi-//cum  ac  Mathematicum.//  (Woodcut  as 
in  the  1542  edition.)  Vvitebergae // ex  ofificina  Haeredvm  // 
Georgii  Rhavv//M.D.LXI.'    (F.  i,  r.) 

Colophon.  '  Vvitebergae  //  ex  officina  Haeredvm  //  Georgii 
Rhavv//M.D.LXI.'    (F.  88,  v.) 


PRINTED    BOOKS  205 

Description.   8°,  9.8  X  14.9  cm.,  the  text  being  "j ."]  x  i  1.2  cm. 
88  ff.  unnumb.,  23  11.    Wittenberg,  1561. 
See  p.  201. 

GEMMA   FRISIUS.     Ed.  pr.  1540.  Paris,  1563. 

See  p.  200. 

Title.  The  title  page  is  substantially  the  same  as  that  of 
1549  (p.  202),  with  the  following  addition:  lOth  line,  '  Quibus 
demum  ab  eodem  Peletario  addita;  funt  Radicis  //  vtriufque 
demonftrationes.' 

Description.  8°,  10.4  X  16.3  cm.,  the  text  being  7.1  x  12.5 
cm.     2  ff.  unnumb.  +  102  numb.  =  104  ff.,  26  11.    Paris,  1563. 

See  p.  201. 

GEMMA  FRISIUS.    Ed.  pr.  1540.  Wittenberg,  1563. 

See  p.  200. 

Title.  '  Arithme-//tic3e  practicae  me-//thodvs  facilis,  per // 
Gemmam  Frifium  Medicum  ac  // Mathematicum.// VVitcbergae 
//M.D.LXIII.'  (F.  I,  r.) 

Colophon.  '  VVitebergae // ex  ofificina  Haeredvm.//Georgii 
Rhavv.// M.D.LXIII.'    (F.  88,  r.) 

Description.  8°,  10. 5  X  i  5.6  cm.,  the  text  being  8.  i  x  11.2  cm. 
88  ff.  unnumb.,  23  11.     Wittenberg,  1563. 

Editions.    See  p.  200. 

GEMMA   FRISIUS.     Ed.  pr.  1540.  Venice,  1567. 

See  p.  200. 

Title.  '  Aritmetica/Zprattica  facilissima,//composta  da  Gem- 
ma Frisio  Medico,// et  Matematico  ;  //  Con  I'aggiunta  dell'  Ab- 
breuiamento  de  i  Rotti  Aftronomici  di  //  Giacomo  Pelletario; 
&  del  conofcere  a  mente  le  Calende,//gr  Idi,  le  None,  le  Fefte 
Mobili,  il  luoco  del  Sole,  &  della  Luna  nel  //  Zodiaco;  &  la  dimo- 
ftratione  della  Radice  Cubica  :  lequali  tutte  //cofe  dal  latino,  ha 
in  quefta  lingua  ridotte  Oratio  Tosca-//nella  della  famiglia  di 
Maeftro    Luca  Fiorentino  ;    &  halle   de-//dicate  //  alio   illvstre 


2o6  RARA   ARITHMETICA 

fignore,  il  fignor/ZEttore  Podocataro.// (Woodcut.)  InVenetia, 
Apreffo  Giouanni  Bariletto.// MDLXVII.'    (F.  i,  r.) 

Colophon.  '  Registro.//  *ABCDEFGHIKL  M.//  Tutti 
fono  Quaderni;  eccetto  M,//che  e  Quinterno  //In  Venetia,// 
Apreffo  Giouanni  Bariletto.//MDLXVII.' 

Description.  4°,  14.9  x  20.7  cm.,  the  text  being  10.6  X  17.4 
cm.    5  ff.  unnumb.  +  5  i  numb.  =  56  ff.,  40-42  11.    Venice,  1567. 

Editions.  See  p.  200.  This  is  the  only  Italian  edition  of  the 
sixteenth  century. 

See  p.  201.  The  '  Maeftro  Luca  Fiorentino '  mentioned  in  the  title 
page  was  a  well-known  arithmetician  of  Florence.  A  manuscript  of  his 
of  c.  1425  is  described  in  the  second  part  of  this  work. 

GEMMA    FRISIUS.     Ed.  pr.  1540.  Cologne,  1571. 

See  p.  200. 

Title.  '  Arithme-//ticae  practicae//methodvs  facilis,  per  Gem- 
mam  //  Frifium,  Medicum  ac  Mathematicum,  iam  re-//cens  ab 
ipfo  authore  emendata,  &  multis // in  locis  infigniter  aucta.// 
Hvc  accesservnt  lacobi  Pe-//letarij  Cenomani  annotationes : 
Eiufdem  item  de // Fractionibus  Aftronomicis  compendium:  Et 
// de  cognofcendis  per  memoriam  Calendis,// Idibus,  Nonis, 
Feftis  mobilibus, // &  loco  Solis  &  Lunae  in// Zodiaco.// Nunc 
vero  a  loanne  Stein  recognita,  &  no-//uis  aucta  additionibus.// 
(Woodcut  with  motto:  Benedices  //  Coronae  Anni//Benignitatis 
//Tvae.  Psal.  64//)  Colonise,// Apud  Maternum  Cholinum.// 
M.  D.  LXXI.//Cum  gratia  &  priuilegio  Caef.  Maieft.'    (P.  i.) 

Description.  8°,  9.5  X  15.8  cm.,  the  text  being  6.5  X  12.  i  cm. 
3  pp.  unnumb.  +  281  numb.  =  284  pp.,  28-30  11.  Cologne,  1571. 

See  p.  201. 

GEMMA    FRISIUS.     Ed.  pr.  1540.  Leipzig,  1575. 

See  p.  200. 

Title.  'Arithme-//ticae  practicae // methodvs  facilis,// per // 
Gemmam  Frisivm  //  Medicum  ac  Mathematicum.//  (Woodcut 
representing  a  counting  house,  with  line  reckoning.)  Lipsiae  // 
lohannes  Rhamba  excudebat.// M.  D.  LXXV.'    (F.  i,  r.) 


PRINTia)    1K)()KS  207 

Description.  8°,  9.5  X  15.3  cm.,  the  text  bcini;-  7.8  x  11  cm. 
87  ff.  unnumb.,  23-24  11.     Leipzig",   1575. 

See  p.  201.  The  book  has  been  rebound  with  added  pages  which 
have  been  used  for  manuscript.  These  notes  are  in  Latin,  in  a  (ier- 
man  sixteenth-century  hand,  and  include  both  work  on  the  fundamental 
operations  and  numerous  commercial  problems. 

GEMMA    FRISIUS.     Ed.  pr.   1540.  Paris,  1578. 

See  p.  200. 

Title.  The  title  page  is  substantially  the  same  as  that  iA 
1542  (p.  201),  except  for  the  following:  '  In  me  Mors,// In  me 
Vita.//Parisiis,//Apud  Hieronymum  de  Marnef,  &  viduam // 
Guliel.  Cauellat,  fub  Pelicano  monte  D.  Hilarij.//  i  578.'   (F.  i ,  r.) 

Description.  8°,  10.9  X  17.3  cm.,  the  text  being  7.4  X  14.  i  cm. 
3  ff.  unnumb.  +  93  numb.  =96  ff.,  29  11.    Paris,  1578. 

See  p.  201. 

GEMMA    FRISIUS.     Ed.  pr.  1540.  Antwerp,  1581. 

See  p.  200. 

Title.  'Arithmeticae//practicae  methodvs  fa-//cilis,  per  Gem- 
mam  Frisivm,  Medi-//cum,  ac  Mathematicum  confcripta  :  iam 
recens  ab  Auctore //pluribus  locis  aucta  &  recognita.// In  ean- 
dem  loannis  Steinii  &  Iacobi//Peletarii  Annotationes.//Eiufdem 
de  Fractionibus  Aftronomicis  Compendium,// &  de  cognofcendis 
per  memoria  Kalendis,  Idibus,  No-//nis,  Feftis  mobilibus,  loco'qj 
Solis  &  Lun^  in  Zodiaco. // (Woodcut  of  astronomer.) // Ant- 
verpiae.// Apud  loannem  Bellerum  ad  infigne  Aquilae //aureae. 
Anno  1581.'    (P.  I.) 

Description.  8°,  9.7  X  15.7  cm.,  the  text  being  6.5  X  13  cm. 
5  pp.  unnumb. -f  178  numb.+  i  blank  =  184  pp.,  3i  11-  Antwerp, 
1 58 1.  With  this  copy  is  bound:  'Ivl.  Pacii//a  Beriga // Insti- 
tvtiones//Logicae,//in  vsvm  scholarvm//Bcrnensivm //editae. 
//Bernse.  M.D.C 

See  p.  201. 


2o8  KARA   ARITHMETICA 

GEMMA    FRISIUS.     Ed.  pr.  1540.  Wittenberg,  1583. 

See  p.  200. 

Title.  '  Arithme-//ticae  Practicae  //  Methodus  facilis,//  per  // 
Gemmam  Frisivm  //  Medicum  ac  Mathematicum.// (Woodcut  as 
in  the  1542  edition.)  Vvitebergae//Ex  officina  MatthseiVVelacij. 
//M.  D.  LXXXIII.'    (F.  I,  r.) 

Description.  8°,  9.5  X  15  cm.,  the  text  being  7.9  X  11. 2  cm. 
87  ff.  unnumb.,  23  11.    Wittenberg,  1583. 

See  p.  201. 

GEMMA  FRISIUS.     Ed.  pr.  1540.  Leipzig,  1588. 

See  p.  200. 

Title.  '  Arithme-Z/ticae  practicae // methodvs  facilis,// per // 
Gemmam  Frisivm // Medicum  ac  Mathematicum.// Lipsiae,// 
Anno  M.  D.  LXXXVIII.'     (F.  i,  r.) 

Coloplwn.  '  Lipsiae,// ex  officina  typo-//graphica  Abrahami  // 
Lambergi.//Anno//M.  D.  LXXXVIII.'     (F.  87,  r.) 

Description.    8°,  9.4  X  15  cm.,  the  text  being  7.8  X  1 1.5  cm. 

87  ff.  unnumb.,  24  11.     Leipzig,  1588. 
See  p.  201. 

GEMMA   FRISIUS.     Ed.  pr.  1540.  Leipzig,  1592. 

See  p.  200. 

Title.  '  Arithme//ticae  Practicae // Methodvs  faciHs,//per // 
Gemmam  Frifium  Medi-//cum  ac  Mathematicum.//  (Woodcut  as 
in  the  I  542  edition.)    Lipsiae,//  Anno  //  M.  D.  XCIL'    (F.  i ,  r.) 

Colophon.  (Woodcut  of  Pegasus.)  '  Lipsiae, //ex  officina  typo-// 
graphiaAbrahami//Lambergi.//Anno//M.D.XCII.'   (F.88,r.) 

Description.    8°,  9.2  X  15.5  cm.,  the  text  being  8.1  X  1 1.4  cm. 

88  ff.  unnumb.,  24  11.     Leipzig,  1592. 
See  p.  201. 

HENRICUS   URANIUS.     Ed.  pr.  1540.  Solingen,  1540. 

A  German  classicist,  born  at  Reesz,  Prussia.    He  lived  at  Emmerich  (Em- 
rich)  when  he  wrote  this  work. 

Title.    See  Fig.  102. 


PRINTED    BOOKS  209 

Description.  8°,  lOX  15  cm.,  the  text  being  7.9  X  11.7  cm. 
20  ff.  unnumb.,  28  11.     Solingcn,  1540. 

Editions.    There  was  no  other  edition. 

This  is  a  semi-historical  discussion  of  the  various  measures  which  the 
sixteenth  century  received  from  the  Roman  civilization,  or  which  were 

NVMARIA,    MENSVRIS  ET  PON- 

dcribus  Epitome  ex  Budaeo.Portio.Alciato 

CT-Georgio  Agricolaconcinnata 

pcrHcnricum  Vranium 

RcITcnfcm. 

IN     Z  O  I  L  V  M. 

OmnUfunt  dodis  ante  probaU  wriV. 
Ocyui  h'mcfugito.fi  ncfcis  quo  fit  emdum: 
EnmonlirAho,  malm  Zoile  adito  crnccm* 


^ALINGIACI,  loannes  Sorer 
cxcudebac.  AN.M.D.XL. 

Fig.  102.   Title  page  ok  ukanius 

mentioned  in  the  most  commonly  read  classics  of  the  Renaissance  jieriod. 
No  arithmetical  operations  are  given  in  the  book.  It  begins  with  the 
fractional  parts  of  the  as,  the  twelfth  being  called  the  uncia  (the  Troy 
ounce),  the  sixth  the  sextans,  the  fourth  the  quadrans,  and  so  on. 


2IO  RARA   ARITHMETICA 

PHILIP  MELANCHTHON.     Ed.  pr.  1540.       Leyden,  1540. 

Born  at  Bretten,  Baden,  February  6, 1497  ;  died  April  19, 1560.  Ilis  family 
name  was  Schwarzerd;  he  assumed  the  Greek  equivalent,  Melanchthon,  when 
he  entered  the  university  of  Heidelberg  (1509).  He  was  one  of  the  most 
famous  classicists  of  the  Renaissance,  and  a  leader  in  the  Reformation. 

Title.    See  Fig.  103. 

M  A  T  H  E  M  A- 

TICARVM    DISCI' 

PLI  N  AR  VM,    T  VM 

ETIAM     ASTROLO- 

C  I  A  E     E  N  C  O- 

M  I  A, 

* 

PER    PHI  I.     MELANCHT. 

ITEM 

Phasnomcnaloachimi  Camcrarn> 

clcgantifsimo  carininc 

defaipta. 


\ 


APVD    SEB.    GRYPHIVM 
L  V  G  D  V  N  I, 

1540- 

Fig.   103.   Title  page  of  melanchthon 


PRINTKI)    IU)()KS  211 

Description.  8°,  9.9  X  15.2  cm.,  llic  Icxl  bcinj;  7.2  x  12.2  cm. 
40  pp.  numb.,  30  11.     Lcydcn,  1540. 

Editions.    There  was  no  other  separate  edition. 

The  work  of  Melanchthon  consists  of  three  letters  :  ( i  )  to  Simon 
(^rynaeus,  dated  'Mtebergte,  mefe  Augnfto.  M.  D.  XXXI ' ;  (2  )  to  Johann 
Reiffenstein,  dated  '  Menfe  Augusto.  Anno  M.  D.  XXXVI' ;  (3)  to  Johann 
Schonerus  (see  p.  178),  dated  '  Vuittebergce,  Mefe  Augusto,  Anno  M.  I). 
XXXVI.'  These  epistles  are  all  upon  the  value  and  the  nature  of  mathe- 
matical thought,  and  are  replete  with  classical  and  religious  references. 
As  mathematics  they  have  no  value. 

The  verses  of  Camerarius,  '  loachimi  Camerarii  Phaenomena  ad 
clarissimum  ivvenem  Danielum  Stibarum,'  relate  chiefly  to  astronomy. 

MAGNUS   AURELIUS   CASSIODORUS. 

Ed.  pr.  1540.  I'aris,  1550. 

("AssioDoRirs.    I'.orn  at  Scylaceum,  c.  470  ;  died,  probably  at  Rome,  c.  564. 
A  Roman  statesman  and  historian. 

Title.  'Aurelij  Cafsiodori  Se-//natoris  Cos.qve  R(>mani//de 
quatuor  Mathematicis  difciplinis  //  Compendium.  //  Parisiis  // Apud 
Vafcofanum,  uia  lacobaea  ad  infigne  Fontis.//  M.  D.  L.'  (F.  i,  r.) 

Description.  4°,  15.9  X  20.8  cm.,  the  text  being  9.7  X  17.2  cm. 
I  f.  unnumb.  +  7  numb.  =  8  ff.,  25-29  11.    Paris,  1550. 

Editions.  Paris,  1540;  ib.,  1550,  4°  (here  described);  ib.,  1580. 
The  'Opera'  appeared  in  Paris  in  1 598 (and  1 584  ?).  The  Compen- 
dium is  embodied  in  the  'Disciplinarum  liberalium  orbis,exP.  Con- 
sentio  et  Magno  Aurelio  Cassiodoro,'  published  at  Basel  in  1528. 

This  brief  treatise  on  the  nature  of  arithmetic,  music,  geometry,  and 
astronomy,  the  four  mathematical  disciplines,  was  held  in  high  esteem 
in  the  Middle  Ages. 

Othertvorks 0/1540.  Anianus,  p.  32,  1488  ;  Beldamandi,  p.  15, 1483  ; 
Benese,  p.  182,1536;  Borghi,  p.  21,  1484;  Ortega,  p.  93,  15  12  ;  Paxi, 
p.  79,  1503  ;  Rudolfif,  p.  152,  1526  ;  p.  160,  1530;  Scheubel,  p.  233, 
1545;  Wolphius,  p.  154,  1527;  Anonymous  (Ortega?,  p.  91,  1512), 
'  CEuvre  tres  subtile  y  profitable  de  I'art  y  science  de  arismeticque  y 
geometric  translate  nouvellement  d'Espaignol  en  Francoys,'  Paris,  8°. 

Works  of  154J.  Agrippa,  p.  167,  1531  ;  Albert,  p.  180,  1534  ;  Cardan, 
P-  193.  1539;  Riese,  p.  139,  1522  ;  Ringelbergius,  p.  167,  1531  ;  Ta- 
gliente,  p.  115,  15  15  ;   Georg  Rheticus,  Arithmetic,  Strasburg. 


212  RARA  ARITHMETICA 

ANONYMOUS.     Various  authors. 

Ed.  pr.  1542.  Cologne,  1542. 

Title.    See  Fig.  104. 

ARITHME    ^ 

TICES   INTRODV-T^'^ 

Slio  ex  uarijs  authoribus  con^ 
cinnata^ 


,^it^*^*"f 


MfCCa 


a^i^JO 


W 


ColonicetxcudehatToannesGymnicui 
K^nm    M.    D.    XLIL 

^'w         i'U»w«p      ^(U^v>t~     ■'Stx^        Cu<U'       wc/yl- 

FiG.  104.   Title  page  of  the   1542  anonymous  work 

Description.    8°,  9.7  X  15.1  cm.,  the  text  being  6.9  X  12.3  cm. 
20  ff.  unnumb.,  25-29  11.    Cologne,  1542. 


PRINTED    HOOKS  213 

Editions.  Cologne,  i  542,  8°  (here  described);  il).,i546;  Dort- 
mund, 1549,  8°  (see  below). 

This  is  one  of  several  anonymous  compilations  made  in  the  sixteenth 
century  for  use  in  the  Latin  schools.  It  has  no  merit,  save  that  of  brevity. 
It  contains  a  brief  treatment  of  the  fundamental  operations,  followed  by 
a  chapter  '  I)e  Progrefione,'  the  '  Regvla  mercatorvm  feu  de  tribus,'  and 
7  pages  '  De  minutijs.'  I  notice  that  folio  B  6,  v.,  is  an  exact  copy  of 
Wolphius  (see  j).  154)  folio  B  i,  v.  This  is  the  same  as  the  work  next 
mentioned,  published  at  Dortmund  in  1549. 

ANONYMOUS.     Ed    pr.  1542.  Dortmund,  1549. 

Title.  '  Brevis // Arithme//ticcs  Intro-//dvctio  ex  Variis // 
Authoribus  con-//cinnata.  Tremoniae  excud.  Melch.  Soter./ 
Anno  M.D.LXIX.'    (P.  i.) 

Description.  8°,  9.8  X  14.9  cm.,  the  text  being  6.1  x  12.3  cm. 
48  pp.  unnumb.  (i  blank),  25  11.    Dortmund,  1549. 

See  above. 

ROBERT   RECORDE.     Ed.  pr.  c.  1542.  London,  1558. 

Born  at  Tenby,  Pembroke,  c.  1510;  died  in  .Southwark  prison,  prolmbly 
.soon  after  June  28,  155S  (the  date  of  his  will).  He  was  educated  at  Oxford 
and  Cambridge,  and  taught  mathematics  at  the  former  and  probably  at  the 
latter  university.  He  became  royal  physician  and  wrote  on  medicine  as  well 
as  mathematics. 

Title.    See  Fig.  105. 

ColopJion.  'Imprinted  at  London  in  Paules  churchyard /Vat 
the  figne  of  the  Brafen  Serpent  //by  Reginalde  Wolfe.// Anno 
Domini  M.  D.  LVIIL'    (F.  205,  v.) 

Description.  8°,  8.9  X  13.  i  cm.,  the  text  being  8  x  11.8  cm. 
205  ff.  unnumb.,  31  11.     London,  1558. 

Editions.  There  is  considerable  uncertainty  as  to  the  date  of 
the  first  edition  of  this  work  of  Recorde's.  It  appeared,  however, 
between  1540  and  1542.  For  a  discussion  of  the  question  see 
the  Dictionary  of  National  Biography,  and  De  Morgan,  p.  22. 
The  former  says  there  were  twenty-seven  editions  of  the  book, 
but  there  were  at  least  twenty-eight  (see  p.  214).  On  account  of 
the  influence  of  the  work  on  English  education,  the  bibliography 


214 


RARA   ARITHMETICA 


has  been  extended  through  the  seventeenth  century.  London, 
c.  1542;  ib.,  1543,8°;  ib.,  1549,8°;  ib.,  1551,  8°;  ib.,  1556; 
1552;  London,  1558,  8° (here  described);  ib.,  1561,  8°  (the  earliest 
seen  by  De  Morgan);  1570;  London,  1571,  8°;  ib.,  1573;  ib., 
1577;  ib.  1579,  8°  (p.  217);  ib.,   1582,  8°  (the  Mellis  edition); 


THE    GROVND     OF 

A  R  T  H  S: 

5r!riTci)(nfftt)Ctt)oojl5Cfinb  pjacfifc  of 

^ii'tfjinetilie,  botf)uiUiJ)olctmml»iC5 

flitD  irflctions  ,  eftcca  moie  eafpcc 

anD  crflctcc  fozte,  tfieii  anyc  Ip1\c 

uev8  ueib  fiOi 

iSIJcDebp  f^.ROBERTB 
R  E  C  O  R  D  E 

2)otfoj  of  ^'^tjpftiie. 


Swiss 


E    PACE    OE    THE     I 5 58    RECOKDE 


ib.,  1586,  8°;  ib.,  1590,  8°  (the  Dee  and  MelHs  edition);  ib., 
1594,  8°  (p.  207);  ib.,  1596,8°  (p.  219);  ib.,  1618,  8°;  ib.,  1623, 
8°;  1636 ;  London,  1646,  8°  (p.  219);  ib.,  c.  1646  (p.  220);  1652; 
1654;  London,  1662,  8°  (p.  220);  ib.,  1668,  8°  (p.  221);  1673; 
1699.    Presumably  all  of  these  were  published  at  London. 


PRINTED    BOOKS  215 


ADDITION. 

none  otbcr  cramplcs  f o?  to  Icnrne  t^e  numc 
ration  cf  tDisfoimc. 

But  tljie  OjaU  vcu  mat!ic,tljat  as  pou  tJfO 
tn  tl3c  otljcc  hrnT)  of  aritljinctift  ict  a  p:ic!sc 
intDcplftCCs  oJ  tUDufauBcs,  mtOisttiooiKc 
vou  iaU  Cct  a  ttarrcas  p  ou  fee  bcfo2c* 

S.II^cn  If,  pcrccauc  ]^umrrattcn :  but  J 
piapc  rou,  bott)  fl?cU  3  bo  in  tins  act  to  dDte 
•tteoo  dimmc?  031  mo;c  togit^er  ? 

ADDITION. 

Maftcr* 

tf^  l)C  eaficCt  toap  lu  tbis  arte,»9  to  arte 

^^  but  ttoo  fummc9  at  ones  togrtbcc: 

t)oto  be  it,  pou  mafc  abbe  mc?e,as  J  U)?l  tc\ 

l-ouanone*   tbcccfojc  tobcnnc  pou  torllc 

abbe  tttio  fummcs,poufl)Ml  fpitte  Ctt  botime 

one  of  tbem,it  fouctb  not  ti)t)icbf ,  onri  tt)cn 

bp  it  biflto  a  li^ne  ccoflie  tt)e  otbcr  IptrtgAlnb 

aftccttjacbe  icttcbounetbeotbcr  fuinmc  >  fo 

t»)attbat  Ipncmapc 

l)CbcttbcnetDcm;a0 

If  pou  tboulbe  atJbt 

Wrion    *^5P  to  834* ,  pou 

muafctpoucCumcd 

a0pourceberc» 

anb  t^cu  <f  pou 
Ipft,  pou  mapc  abDr 

tt)c  one  totl)C  otber  m  tljc  iV.mr  place.  0:  els 
poumapatbf tDmDottjrroDitDccin  amXo 
place :  tobici)  ^ar,bpcaurc  ins  mod: plpiica 

Fig.  106.    Counter  reckoning  eroim  the   1558  kecokue 


ttooo 

mnned 


2i6  RARA   ARITHMETICA 

The  first  arithmetic  to  be  published  in  England  was  that  of  Tonstall 
(p.  132).  There  must  have  been  several  other  books  published  between 
that  date  and  the  first  appearance  of  Recorde's,  because  in  the  preface 
to  the  latter  the  author  says  :  '  And  if  any  man  obiect,  that  other  books 
haue  bene  written  of  Arithmetike  alreadie  fo  fi.ifiiciently,  that  I  needed 
not  now  to  put  penne  to  the  booke,  except  I  wil  codemne  other  mens 
writings  :  to  them  I  anfwere.  That  as  I  codeme  no  mans  diligence,  fo 
I  know  that  no  man  can  fatisfie  euery  man,  and  therefore  like  as  many 
do  efteeme  greatly  other  bookes,  fo  I  doubt  not  but  fome  will  like  this 
my  booke  aboue  any  other  Englifh  Arithmetike  hitherto  written,  &  namely 
fuch  as  fhal  lacke  inftructers,  for  whofe  fake  I  haue  plain-ly  fet  forth  the 
exaples,  as  no  book  (that  I  haue  feene)  hath  hitherto :  which  thing  fhall 
be  great  cafe  to  the  rude  readers.'    (From  the  1594  edition.) 

This  is,  however,  the  first  commercial  arithmetic  of  any  note  used  in 
the  English  schools.  It  is  written  in  the  form  of  a  dialogue  between 
the  master  and  his  pupil,  and  the  language  is  so  formal  that  it  seems 
strange  that  the  book  should  have  been  so  successful.  The  first  part 
is  devoted  to  integers,  the  fundamental  operations  being  followed  by  a 
section  on  denominate  numbers.  This  is  followed  by  proportion  and 
the  'golden  rule '  of  three,  the  backer  rule  of  three  (inverse  proportion), 
the  double  rule  of  three  (compound  proportion),  the  rule  of  three  com- 
posed of  five  numbers,  and  the  rule  of  fellowship  (partnership).  The 
second  part  relates  to  fractions,  and  includes  the  same  general  topics 
as  the  first,  together  with  alligation  and  the  rule  of  false.  Counter  reck- 
oning is  given  (Fig.  106)  as  well  as  computation  with  Arabic  numerals. 

Among  other  works,  Recorde  wrote  '  The  Castle  of  Knowledge '  (p. 
253)  and  'The  Whetstone  of  Witte '  (p.  286),  the  latter  being  chiefly 
on  algebra.  His  geometry,  '  The  Pathway  to  Knowledge,'  appeared  in 
London  in  155 1,  4°;  ib.,  1574,  4°. 

Recorde's  works  were  the  most  influential  English  mathematical  pub- 
lications of  the  sixteenth  century.  Thomas  Willsford,  in  his  1662  edition 
of  the  '  Ground  of  Artes,'  was  able  to  say  with  much  truth  that  this  book 
was  '  entail'd  upon  the  People,  ratified  and  fign'd  by  the  approbation  of 
Time.' 

Of/ier  70orks  0/1^42.  Albert,  p.  180,  1534;  Cardan,  p.  193,  1539  ; 
Finaeus,  p.  163,  1530-32  ;  Gemma,  p.  200,  J540;  Ortega,  p.  93,  15 12  ; 
Diego  el  Castillo,  'Tratado  de  quentos,'  Salamanca,  4°;  Giambattista 
Verini,  '  Spechio  del  mercatanti,'  Milan,  8°  (Brunet  says,  '  Libro  de 
Abaco  e  gioco  de  memorie,'  Milan,  sm.  8°);  Han  vanderWehn,  '  Exem- 
pelrechenschaft  der  Kegel  de  Tri,  die  man  nennt  die  Kaufmanns  glildene 
Kegel  ganz  und  gebrochen,'  s.  1.,  8°. 


PRINTICI)    I'.OOKS  217 

ROBERT   RECORDE.     Ed.  pr.  c.   i  542.  Eoiulon,   1579. 

See  p.  2  1 3. 

Title.  '  The //Groundc  of  Artes  :// tcachin^^  the  work  and 
pra-//ctife  of  Arithmetikc,  bothe  in  whole  numbers //and  Erac- 
tions,  after  a  more  eafyer  and  //  exacter  forte  than  any  like  hath 
hither-//to  bin  fet  foorthe  ://  Made  by  Mayfter  Roberte  Re-// 
cord,  Doctor  in  I^hyfike,  and  now  of  late//  diligently  ouerfeene 
and  augmented  //  with  new  and  neceffarie  //Additions.// 

I.  D. 

That  which  my  freende  hath  well  begonne, 
For  very  loue  to  common  weale, 
Neede  not  all  whole  to  be  new  done, 
But  new  encreafe  I  do  reueale. 

Some  thyng  heerein,  I  once  redreft, 
And  nowe  agayne  for  thy  behoofe, 
Of  zeale  I  doe,  and  at  requeft, 
Both  mend  and  adde,  fitte  for  all  proofe. 

Of  Numbers  vfe,  the  endleffe  might, 
No  witte  nor  language  can  expreffe, 
Applie  and  Trie,  both  day  and  night. 
And  then  this  truth  thou  wilt  confeffe. 

Printed  at  London  by  H.  Bynneman.// Anno  Domini.  1579.' 
(E.  i,r.) 

ColopJion.  '  Imprinted  at  London  by  Henry // Binneman,  and 
John  Harifon.//Anno  Domini  M.D.LXXVIL'     (E.  261,  r.) 

Description.    8°,  8.7  X  13.9  cm.,  the  text  being  6.5  X  1 1.7  cm. 

261  ff.  unnumb.,  27  11.     London,  1579. 

It  will  be  noticed  that  the  colophon  is  dated  1577.  This  is  therefore  one  of 
the  cases  where  a  large  edition  was  printed,  and  a  new  title  page  was  added  from 
year  to  year  as  necessary.     The  edition  is  more  rare  than  its  date  would  suggest. 

ROBERT  RECORDE.     Ed.  pr.  c.  1542.  London,  1594. 

See  p.  213. 

Title.  '  The  //  Grovnd  of  //  Artes,  teaching  //  the  perfect 
worke  and  practife  //of  Arithmeticke,  both  in  whole  numbers  // 
and  Eractions,  after  a  more  eafie  and  exact  //fort,  than  hitherto 


2i8  RARA   ARITHMETICA 

hath  been  fet  foorthe.// Made  by  M.  Robert  Record,// D.  in 
Phifickey/And  now  lately  diligently  corrected  and  beauti-//fied 
with  fundry  new  Rules  and  neceffary  Addi-//tions  :  And  further 
endowed  with  a  third  part,  of  //  Rules  of  Practife,  abridged  into 
a  briefer  method  //  than  hitherto  hath  bene  publifhed  :  with 
di-//uerfe  fuch  neceffairie  Rules  as  are // incident  to  the  trade 
of // Merchandife.// Whereunto  are  alfo  added  diuerfe  Tables 
and  In-//ftructions  that  will  bring  great  profite  and  delight  vnto 
Merchants,  Gentlemen,  and  others,// as  by  the  Contents  of  this 
Trea-//tife  fhall  appeare.//  By  lohn  Mellis.//  At  London,// 
Imprinted  by  T.D.  for  lohn  Harifon,  at  the  Greyhound  in// 
Paules  Churchyard.//  1594.'// (P.   i.) 

Colophon.  '  Imprinted  at  London  by  Thomas  Davvfon,  for 
lohn // Harrifon,  dwelHng  in  Paules // Churchyard,  at  the  figne 
of //the  Greyhound,// 1594.'    (P.  558.) 

Description.  8°,  10.  i  X  15.5  cm.,  the  text  being  j.Gx  13.1 
cm.  34  pp.  unnumb.  +  493  numb.  33-525  +  i  blank  =  528  pp., 
30-31  11.     London,  1594. 

Editions.  See  p.  213.  This  is  probably  the  third  edition  by 
Mellis.  In  the  dedication  he  says  :  '  And  feeing  that  within  this 
8.  yeares,  two  impreffions  of  thefe  my  labors  dedicated  to  your 
Worfhip  are  already  worne  out .  .  .',  although  not  much  reliance 
can  be  placed  on  the  statement,  since  it  also  appears  in  the  1596 
edition. 

See  p.  216.  The  first  two  parts,  covering  404  pp.  of  this  book,  are 
substantially  identical  with  the  1558  edition  (p.  213).  The  third  part 
(pp.  405-557),  the  work  of  John  Mellis,  also  appeared  in  the  1591 
edition.  It  includes  the  '  Rules  of  Practife,'  '  The  order  &  worke  of  the 
Rule  of  three  in  broken  Numbers,  after  the  trade  of  Marchants,  digreff- 
ing  fomething  from  M.  Recordes,' '  Loffe  and  Gaine,'  *  Rules  of  Payment' 
(equation  of  payments),  barter,  exchange,  interest,  and  other  business 
applications,  together  with  a  chapter  on  '  Sportes  and  Paftimes  done  by 
Number.'  These  mathematical  recreations  had  already  appeared  in 
printed  textbooks,  and  they  played  an  interesting  role  until  the  latter 
half  of  the  nineteenth  century.  Those  in  this  treatise  related  to  number 
guessing,  the  rules  being  easily  developed  by  our  present  algebra,  but 
rather  mysterious  by  sixteenth-century  arithmetic. 


PRINTi:!)    1U)(JKS  219 

ROBERT  RECORDE.     Ed.  pr.  c.   1 542.  London,  1596. 

Seep.  213. 

Title.    The  title  pa<;e  is  missing. 

Colophon.  '  Imprinted  at  London  by  Richard  //  l^^ield,  for  lohn 
Harrifon,  dwelling //in  Pater  nofter  Row  at  the // figne  of  the 
Greyhound.// 1596.'    (P.  559) 

Description.  8°,  9.8  X  15.5  cm.,  the  text  being  8  X  13.3  cm. 
559  pp.,  30-31  11.    London,  1596. 

Editions.  See  p.  213.  A  note  on  the  last  page  says  that  this 
copy  was  bought  in  1686  for  is.  6^.,  not  a  very  low  price  at  that 
time  for  a  book  only  ninety  years  old.  This  edition  is  practically 
identical  with  that  of  1594  (p.  217). 

ROBERT  RECORDE.     Ed.  pr.  c.  1542.  London,  1646. 

Seep.  213. 
Title.  '  Records  Arithmetick  ://  or,//  The  Grovnd  //  of  Arts  : 
//  Teaching  //  The  perfect  work  and  Practice  of  Arithmetick,// 
both  in  whole  Numbers  and  Fractions,  after  a  more//earie  and 
exact  form  than  in  former  time  hath  been  fet  forth  ://Made  by 
M.  Robert  Record,  D.  in  Phyfick. //Afterward,  augmented 
by  M.  John  Dee.// And  fmce  enlarged  with  a  third  part  of  Rules 
of  Pra-//ctife,  abridged  into  a  briefer  method  than  hitherto  hath 
been//publifhed,  with  divers  neccffary  Rules  incident  to  the 
Trade //of  Merchandife  :  with  Tables  of  the  valuation  of  all 
Coyns,//as  they  are  currant  at  this  prefent  time.// By  John 
Mellis.//And  now  diligently  perufed,  corrected,  illuftrated  and 
en-//larged;  with  an  Appendix  of  figurate  Numbers,  and  the 
Extraction  //of  their  Roots,  according  to  the  method  of  Chriftian 
Vrftitius  :  with  //Tables  of  Board  and  Timber  meafure  ;  and  new 
Tables  of  Intereft  //upon  Intereft,  after  10  and  8  per  100  ;  with 
the  true  value  of  //  Annuities  to  be  bought  or  fold  prefent, 
Refpited,  or  in  ReverV/fion  :  the  firft  calculated  by  R.  C.  but 
corrected,  and  the //latter  diligently  calculated  by  Rob:  Hart- 
well,  Philomathemat.//Scientia  non  habet  inimicum  nifi  ignoran- 
tem.'//  Fide. fed -  -  Vide.//  London,//  Printed    by 


220  RARA   ARITHMETICA 

M.  F.  for  John  Harifon,  and  are  to  be  fold  by //Geo:  Whitting- 
ton,  and  Nath:  Brooks,  at  the  fign  of // the  Angell  in  Corn-hill. 
1646.'     (P.  I.) 

Description.  8°,  10  X  15.5  cm.,  the  text  being  8.6  X  13.7  cm. 
27  pp.  unnumb.  +  629  numb.  =  656  pp.,  31  11.     London,  1646. 

Editions.    See  p.  213. 

Like  the  1594  edition  this  has  the  additional  'third  part'  by  John 
Mellis.  The  Hartwell  chapter  on  roots  begins  on  p.  573,  and  is  based, 
as  the  title  says,  on  the  work  of  Urstisius. 

ROBERT  RECORDE.     Ed.  pr.  c.  1542.       London,  1646  (?). 

See  p.  213. 

Title.  This  edition  is  said  to  have  been  published  in  1646, 
but  no  date  appears  in  the  book  itself.  It  has  the  same  title  page 
as  the  one  dated  1646,  except  as  follows  :  The  words  '  Records 
Arithmetick'  (l.i)  are  here  omitted  ;  the  last  lines  read  '  Printed 
by  M.  F.  for  John  Harifon,  and  are  to  be  //  fold  at  his  Shop  in 
Pauls-Church-yard.'  The  number  of  words  on  some  of  the  lines 
in  the  title  page  is  different,  but  with  the  above  exceptions  the 
page  is  the  same.  The  body  of  the  book  is  from  the  same  setting 
of  type  as  in  the  other  edition  of  1646. 

Description.  8°,  10. 2  x  15.7  cm.,  the  text  being  8.6  X  13.7 
cm.  27  pp.  unnumb.  +  629  numb.  =  656  pp.,  32  11.  London,  s.  a. 
(1646.?). 

See  p.  216. 

ROBERT  RECORDE.     Ed.  pr.  c.  1542.  London,  1662. 

Seep.  213. 

Title.  The  title  page  is  practically  the  same  as  that  of  1646, 
except  as  to  the  imprint :  '  Printed  by  James  Flefher,  and  are  to 
be  fold  by  Jofeph  //  Cranford,  at  the  figne  of  the  Gunn  in  St. 
Pauls  //  Church-yard.     1662.' 

Description.  8°,  10. 7  X  16.5  cm.,  the  text  being  9.2  x  13.9 
cm.   22  pp.  unnumb. +  536  numb.  =  558  pp.,  33  11.    London,  1662. 

See  p.  216. 


PRINTia)    lUJOKS 


22  I 


ROBERT  RECORDE.     Ed.  pr.  c.  1 542.  London,  1668. 

See  p.  213. 

Title.  The  title  page  of  this  edition  is  practically  the  same  as 
that  of  1646,  except  as  to  the  imprint :  'Printed  by  James  Fief  her, 
and  are  to  be  fold  by  //  Robert  Boulter,  at  the  Turks-head  in 
Bishopsgate-//ftreet,  next  the  great  James.     1668.' 

Description.  8°,  10.4  X  16.4  cm.,  the  text  being  9.2  X  13.7  cm. 
22  pp.  unnumb.  +  536  numb.  =  558  pp.,  33  11.     L(jndon,   1668. 

JOIIANN    FREY.     Ed.  pr.  1543.  Nurnberg,  s.  a.  (1543). 

A  Nurnberg  gauger  of  the  middle  of  the  sixteenth  century. 

Title.    See  Fig.  107. 

Colophon.  *  Gedruckt  zu  Nurnberg  durch//Georg  Wachter.' 
(F.  36,  r.) 

Description.  8°,  9.5  x  15.5  cm.,  the  text  being  7.1  x  10.9  cm. 
36  ff.  unnumb.,  21-26  11.    Nurnberg,  s.  a.  (1543). 

Editions.  There  was  no  other  edition.  The  book  bears  no 
date  except  in  the  dedicatory  epistle  to  the  reader,  which  closes 
with  the  words,  'Anno  1543.'  This  epistle  gives,  also,  the  only 
reference  to  the  author,  who  there  speaks  of  himself  as  '  ich 
Johaii  Frey/ burger  zu  Nurmberg.' 

The  subject  of  gauging  occupied  a  great  deal  of  attention  on  the 
part  of  German  writers  on  arithmetic  in  the  sixteenth  century,  and 
occasionally,  as  in  this  instance,  separate  books  were  prepared.  It  was 
not  so  common  in  England  as  on  the  Continent,  not  appearing,  for 
example,  in  as  extensive  a  work  as  the  Dee  and  Mellis  edition  of  Recorde 
mentioned  above.  It  had,  however,  some  standing  in  the  early  Amer- 
ican arithmetics,  and  is  found  as  a  separate  chapter  as  late  as  the  mid- 
dle of  the  nineteenth  century.  The  title  page  gives  some  idea  of  the 
work  of  the  gauger  before  the  days  of  standardization  of  casks.  The 
American  rule  for  gauging  a  cask  was  substantially  as  follows  :  Add  to 
the  head  diameter  0.7,  0.65,  0.6,  or  0.55  of  the  difference  between  the 
head  and  bung  diameters  (according  to  the  degree  of  curvature  of  the 
staves),  and  multiply  the  square  of  this  sum  by  the  length  ;  divide  by 
359  for  ale  or  beer  gallons,  and  by  294  for  wine.  Thus  a  cask  with 
bung  diameter  36  in.,  head  diameter  30  in.,  and  length  48  in.,  contains 
153-65  ale  gallons. 


222 


RARA   ARITHMETICA 


This  particular  work  is  interesting  because,  although  it  was  printed 
in  1543,  some  of  the  numerals  are  quite  like  those  of  a  hundred  years 
earlier.   The  mediaeval  4,  5,  and  7  are  exclusively  used  in  the  engraved 

£\nnmWkr  McUcini 

wclcbee  inn^dvwic  man  X>md)  t>cn  itLviat>utttn 

anffc^ms'^c^en  \(xnt>6  ^ich/fin  Kitten  3fibc 

rettcrt/vnt>4mii>yethcb6Viibclf4ntevag 

Vifievm/vnty  (olci)e6  mnbrtlt  crhm 

nen  f<^l/2(uff6»tcw  gebeflcrt 

vnt>  gem'ert. 


Fig.  107.    TriLE  page  of  frey 

figures,  although  the  types  used  in  the  body  of  the  book  give  the  later 
forms.  These  mediaeval  figures  may  be  seen  in  several  illustrations  in 
the  second  part  of  this  bibliography. 


PRINTED    HOURS  223 

ANONYMOUS.     Ed.  pr.  1543.  Paris,  1543. 

The  author  of  the  'I'lieoloi^ouiucna  is  iinkiniwn.  He  lived  after  Anatolius 
(Bishop  of  Laodicea,  270  a. D.)  and  probably  after  lamblichus  (fourth  century). 

Title.    Sec  Fig.  108. 

Description.  4°,  15.2  x  23  cm.,  the  text  being  9.1  x  18.1  cm. 
7  pp.  unnunib.  +  6i  numb.  (5-65)4-2  blank  =  70  pp.,  30  11.  Paris, 
1543.  Greek  te.xt,  except  the  dedicatory  epistle,  which,  being 
dated  'Lutitiae  Parifiorum  1543.  6.  Calendas  lulias,'  shows  this 
to  be  the  first  edition. 

Editions.  This  edition  was  unknown  to  De  Morgan.  The 
best  edition  is  that  of  Fr.  Ast,  Leipzig,  18 17. 

A  work  of  little  importance,  on  the  (ireek  theory  of  numbers.  Gow 
(p.  88)  describes  it  as  a  '  curious  farrago.'  Cantor  {Geschichte  dcr  Mathc- 
nmtik,  Kap.  22)  says  that  the  author  may  have  drawn  on  lamblichus. 
The  most  valuable  feature  of  the  work  is  the  light  which  it  throws  on 
an  earlier  work  by  Speusippus,  nephew  of  Plato. 

Other  works  of  i^4.j.  Archimedes,  p.  228,  1544;  Gemma,  p.  200, 
1540;  Glareanus,  p.  192,  1539;  Recorde,  p.  214,  c.  1542;  Regius, 
p.  182,  1536  ;  Sfortunati,  p.  174,  1534  ;  Tonstall,  p.  134,  1522  ;  Nicolaus 
Medlerus,  '  Rudimenta  Arithmeticae  practicae,'  s.  1.,  8°  (with  subsequent 
editions:  Wittenberg,  1550;  ib.,  1558;  Leipzig,  1556;  AV'eissenfels, 
1564,  8°). 

MICHAEL   STIFEL.     Ed.  pr.  1544.  Nurnbcrg,  1544. 

Stiefel,  Stvfel.  Born  at  P^sslingen,  April  19.  1487;  died  at  Jena.  April 
19,  1567.  He  was  a  priest,  a  reformer,  and  a  fanatic,  but  was  one  of  the  most 
sivillful  arithmeticians  of  his  time. 

Title.    See  Fig.  109. 

Colophon.  'Excudebatur  Norimberga; //apud  loh.  Petreium.' 
(F.  325,  r.) 

Description.  4°,  15.5  X  20.2  cm.,  the  text  being  10.2  x  15.1 
cm.     326  ff.  (I  blank,  6  unnumb.),  33  11.     Niirnberg,  1544. 

Editions.  Stifel  wrote  five  works  on  mathematics,  all  but  one 
appearing  after  his  period  of  religious  fanaticism.  These  works, 
with  their  various  editions,  are  as  follows  : 

I.  '  Ein  Rechen  Biichlein  vom  End  Christ.  Apocalysis  in 
Apocalysim,'  Wittenberg,  1532.  A  little-known  work  on  the 
theory  and  mysticism  of  numbers. 


^::. 


i^TA  0EOAO- 

rorMENA      THX    APlGMHTIKHr. 

Habes  hie  o  ffcudio- 

SE    LECTOR,  NOVVM    OPVS- 

culum  antehac  nufquam  cxcufum  ,  in  quo  ita 

Numerorum  ratio  explicatur,vt  non  fit  obfcu- 

rum  intelligere  hanc  arithmetica  ad  intenore 

lUa  de  philofophia  difputationem,quam 

Theologiam  veteres  vocabant, 

conferre  piurimum. 


P  ARISIIS. 

ApudChriftianumwecIielum  fiibfcuto  Bafi- 

lienfi,in  vice  lacob;Eo:&  fub  Pegafo,in  vi- 

coBellouacenfi.        M.  D.  XLIIL 

Fig.   io8.    Title  page  of  the   Theologoumctia 


rRlNTi:!)    liOOKS 


225 


ARITHMETI 

CA    INTEGRA. 

Authorc  Michaek  Stifelio* 


NorimbcrgJC  apud  lohan.Pctrcium. 
AnnoChrifli  m»d.  XLiiiu 

Cum  gratia  Sc  priuilcgio  Ca-farco 
atc^  Regio  ad  Scxenniiim* 

Fl(i.    109.     'riTLE    PAGE    OF    STIFEI.'s    SECOXD    AKITHMETU- 


226  RARA   ARITHMETICA 

2.  *  Arithmetica  Integra,'  Niirnberg,  1544,  4°  (here  described); 
ib.,  1545,  4°;    1546;  Niirnberg,  1548;  ib.,  1586,  4°. 

3.  '  Deutsche  Arithmetica '  (p.  231),  Niirnberg,  1545,  4°- 

4.  '  Rechenbuch  von  der  Welfchen  vnd  Deutfchen  Practick/ 
auff  allerley  vorteyl  vnd  behendigkeit/  mit  erklerung  viler  Exem- 
peln/  .  .  .',  Niirnberg,  1546. 

5.  'DieCoss',Konigsberg,  1553-54,  4°(P-259);  ^S7^  ;  1615. 

The  dedication  and  Melanchthon's  preface,  both  of  which  have  been 
removed  from  this  copy,  are  dated  'Vuitebergse  1543.'  This  copy  has 
evidently  been  owned  by  some  one  unsympathetic  with  Melanchthon, 
because  not  only  has  the  preface  been  removed  but  the  reformer's  name 
has  been  crossed  out  of  the  title  page  (Fig.  109).  The  work  is  one  of 
the  most  scholarly  arithmetics  and  algebras  that  came  out  in  Germany 
in  the  sixteenth  century.  It  is  divided  into  four  books,  the  first  being 
an  arithmetic  and  giving  not  only  the  theory  of  the  subject,  but  a  con- 
siderable amount  of  practical  work.  The  second  book  is  on  irrational 
numbers,  a  chapter  which  we  now  insert  in  algebra,  and  the  third  is 
on  algebra  itself,  the  name  of  which  subject  is  said  to  be  '  a  Gebro 
Aftronomo,  auctore  eius,'  a  common  opinion  at  that  time.  This  work 
did  for  Germany  Avhat  Cardan's  and  Tartaglia's  did  for  Italy.  It  was 
a  storehouse  from  which  subsequent  writers  drew,  and,  although  not  a 
practical  mercantile  book,  it  materially  influenced  even  the  elementary 
textbook  makers.  Stifel  himself  recognized  the  demand  for  such  a  work, 
for  he  says  :  '  Quanqj  autem  plurimi  de  Arithmetica  libelli  extent,  & 
quotidie  plures  noui  gignunter,  ego  tamen  adhuc  nullum  uidi  qui  Integra 
artem  traderet.'    Stifel  makes  much  use  of  the  plus  and  minus  signs. 

ARCHIMEDES.     Ed.  pr.  1544.  Basel,  1544. 

Born  at  Syracuse,  c.  287  B.C. ;  died  there  in  212  B.C.    The  greatest  of  Greek 
mathematical  physicists. 

Ttt/c.    See  Fig.  1 10. 

Colophon.  'Basileae,  per  loannem  //  Hervagivm,  anno  ab  orbe 
re-//dempto,  M.D.  XLIIII.  menfeMartio.'  (P.  69  of  the  last  part.) 

Description.  Fol.,  21.8  X  3  1.7  cm.,  the  text  being  1 3.2  X  23.5 
cm.  455  pp.  (7  blank  and  12  unnumb.),  51  11.  The  work  is 
made  up  of  four  parts,  all  of  the  same  date,  separately  paged, 
and  bound  together.  The  '■^a/u./AiV?;'?,  De  harenae  numero,'  is  the 
sixth  work  in  the  collection.     Basel,  1544. 


PRINTia)    liOOKS 


227 


APXIMHAOYX 

TOY    ZYPAKOYSIOV-,    TA    MEXPI 

ARCHIMEDIS     SYRACVSANI 

PHILOSOPHI    i^C    G  E  O  M  ET  P^^^E     EX. 

cellentidimi  Opera ,  qusequidem extant,omnia,multis I'am  fcculisdefi' 

derata,at(j  a  quam  paucifTimisliacilcnus  uifa,nunc9 

primum  ^  Gra:ce  &.'  Latine  in  lu- 

ccm  cdita. 

Quorum  Catalogum  uerfa  pagi'na  rcperies. 


EVTOCII    ^SC^LONir^E 

IN      EOSDEM     ARCHIMEDIS       LI*- 

bros  Connmentan'a,item  Greece  5<;  Latine, 
nunquam  antea  excu(a. 


Qum  C^J.  Mate  fi. gratia  ((J'prmk^io 
<td  (Quinquennium. 

B  f^  S  I  L  E  <^E, 
Joannes  hleruapim  exaidificit 
An.  M  D  X  L  1  I  I  I . 
Fig.  110.    Title  page  uf  the   1544  akchi.medes 


228  KARA   ARITHMETICA 

Editions.  This  seems  to  be  the  first  edition  of  the  works  of 
Archimedes  to  contain  the  '  De  arenae  numero.'  This  chapter 
also  appears  in  the  1558  edition  (below),  and  separately  in  Paris 
in  1557,  8°  (the  Hamellius  edition,  below).  It  was  not  in  the 
Tartaglia  edition  of  1543.  It  appeared  in  the  elaborate  editions 
of  Barrow  (1675),  Torelli  (1792),  Peyrard  (1807-8),  and  Hei- 
berg  (1880).  Riccardi  says  that  it  '  f u  illustrato  dal  Clavio  ne' 
suoi  commenti  alia  sfera  del  Sacrobosco.' 

There  are  several  works  of  Archimedes  extant.  The  '  De  arenae 
numero,'  included  in  this  edition,  is  not  an  arithmetic,  but  it  treats  of 
the  numeration  of  large  numbers.  It  is  addressed  to  Gelon,  King  of 
Syracuse,  and  proposes  to  show,  '  by  geometric  proofs  which  you  can 
follow,  that  the  numbers  which  have  been  named  by  us  and  are  included 
in  my  letter  to  Zeuxippus  are  sufficient  to  exceed  not  only  the  number 
of  a  sand-heap  as  large  as  the  whole  earth,  but  one  as  large  as  the 
universe.'  Archimedes  then  proceeds  to  develop  a  system  of  numera- 
tion by  octads.  In  this  work  he  incidentally  refers  to  a  fact  which 
would  now  be  expressed  by  the  symbols  x™  •  x"  =  x'"  +  ", 

ARCHIMEDES.     Ed.  pr.  1544.  Paris,  1557. 

See  p.  226. 

Title.  'Paschasii//Hamellii  Regii  // Mathematici//Commen- 
tarius  //  in  //  Archimedis  //  Syracufani  praeclari  Mathematici 
libru  //^de  numero  arenae,  multis  locis  per  //eundem  Hamellium 
//emendatum.//  Lvtetiae  // Apud  Gulielmum  Cauellat,  fub  pin- 
gui  Gallina,//ex  aduerfo  collegij  Cameracenfis.// 1 557-'    ^-  ^^^■) 

Description.  8°,  11  X  17.1  cm.,  the  text  being  7.1  X  12.6  cm. 
48  ff.  numb.,  23-30  11.     Paris,  1557. 

Editions.    See  above. 

This  commentary  of  Hamellius  is  quite  as  satisfactory  as  any  of  the 
earlier  ones. 

ARCHIMEDES.     Ed.  pr.  1544-  Venice,  1558. 

See  p.  226. 

Title.  '  Archimedis  //opera  non  nvlla  //  a  Federico  Comman- 
dino  //  Vrbinate  //  nvper  in  Latinvm  conversa,  //  et  commenta- 
riis//illvstrata.// Quorum  nomina  in  fequenti  pagina  leguntur. 


PRIN'ri-I)    HOOKS  229 

//Cvni  pi"i\ilci;'i()  in  annos  X.// Vonetiis,//a])U(l  I'aiiluin  Mami- 
tium,  Aldi  F.//M  D  LVIII.'    (F.  2,  r.) 

Description.  Fol.,  21.  i  x  30  cm.,  the  text  bcin^^  13.2  X  22.8 
cm.  128  ff.  (2  blank,  8  unnunib.),  39  11.  Venice,  1558.  With 
this  is  bound  the  '  Commentarii // in  oj^cra  non  Nvlla // Archi- 
medis.// Venetiis,// apud  I'aulum  Manutium,  yVldi  F.// M  D 
LVIII.'  (F.  60,  r.)  It  contains  the  book  entitled  '  Archimedis // 
Liber  de  Arence // nvmero.' 

See  p.  228. 

JUAN    SARAVIA,  de  la  calle  Beronese. 

Ed.  pr.  1544.  Medina,  1544. 

A  Spanish  arithmetician,  of  Medina,  of  the  middle  of  the  si.xteenth  century. 

Title.    See  Fig.   1 1  i . 

Colophon.  '  CFue  imprelTa  la  prcfente:  //  obrallamada  Inftrucio 
de  mercaderes  enla  muy// noble  villa  de  medina  d"l  campo  per 
Pedro  de  //  Caftro  Ipreffor.  Acofta  d' Antoiio  de  vrueiia  //  mer- 
cader  d'libros.  Acabofe  atreynta  dias  //  del  mes  de  Julio.  Ano 
de  mil  2  quinie-//tos  3  quarenta  2  quatro  anos.'    (F.  cvij,  i.) 

Descriptioji.  8°,  14x20.1  cm.,  the  text  being  12.1  X  16.9 
cm.  2  ff.  unnumb.  +  105  numb.  (Roman  numerals)  =  107  ff., 
30  11.    Medina,  1544. 

Editions.  Medina,  1544,  4°  (here  described);  ib.,  1547,  4°; 
Venice,  1561,  8°  (p.  231.) 

This  rare  work  was  evidently  written  about  1542  (see  f.  xcviij,  r.). 
It  is  not  a  textbook  on  arithmetic,  but  it  relates  to  commercial  problems, 
the  topics  being  curiously  interspersed  with  biblical  illustrations  to  show 
the  justice  of  the  customs  involved. 

Other  zvorks  0/1544.  Apianus,  p.  155,  1527  ;  Breda,  p.  131,  152  i  ; 
Finaeus,  p.  163,  1530-32  ;  Gemma,  p.  200,  1540  ;  Cirammateus,  p.  123, 
1518;  Kobel,  p.  102,  1514;  Noviomagus,  p.  195,  1539  ;  Peurbach,  p.  53, 
1492  ;  Riese,  p.  139,  1522  ;  Sfortunati,  p.  177,  1534;  Tonstall,  p.  136, 
1522  ;  Vander  Hoecke,  p.  183,  1537  ;  H.  Bock,  'Kin  new  Rechenbuch- 
lein,'  Niirnberg,  8° ;  Johannes  Bogardus,  a  work  on  finger  reckoning 
based  on  Aventinus  (p.  136),  Paris;  Leonard  Hegelin,  'Kin  kimstlich 
Rechenbiichlin  auff   Zyffer   vnnd   andern    hii])schen   Regeln,'   rim(?); 


Jlitunf^bf^^  CfjnxOJ^i^h 


PROPTEWEmTSliVVET^ 


IW-3I:igDAQjaia-\\IVI3IX^ 


C^nftmcionoemcrcaderes 

muy  pJOUccboHi^sSnla  qual  fe  cnfenacomo  bcuen  fo> 
iiicrcadcrestractar^-^yDcqiiemiincnifebiinDeeui- 
tar  Usvfuras^e  todooFostractos  beventcTG-:  com 
p2aef21(TiiiIoconmdocomoaloadcfanrado:ralofm 
dOfy^clascott1p23sDclcen^03Iqm'C3n)^r^acto8c)c 
compama:yotrosnnicbo8Contr£itc£?f  iB^ardciilur 
inenfcreb:Jb!a6el  tvzcto  Dclaclamu^  Z'^rnbicnay 
otrotractado£>ecimbioD»  ignclqiiiilfetrsctit  bdos 
ombios  Hci'tos  yrepjouados/Ulncnamcntcconi^ 
pnc^lopo2cl^ocro20arv^maOeIac^,l(e?i>ero^c^t^ 


Fig.  III.   Title  page  of  saravia 


PRINTED    HOOKS  231 

Antonio  Martin,  'Tractado  de  Arithm^tica  y  Geometria,'  AlcalA,  4°; 
H.  Vuelpius,  '  l)c  minutiis  phyficis  et  practicis  aftronomicae  arithmetit:ae 
regulis,'  Cologne,  4°,  and  *  Libellus  de  communibus  ct  ufitatis  arithnie- 
ticae  practicae  regulis,'  ib.,  4°. 

JUAN    SARAVIA.     Ed.  pr.  1544.  Venice,  1561. 

See  p.  229. 

Title.  'Institvtione//de'  Mercanti//chc  tratta  del  compararc 
//et  vendere,//et  della  vsvra  chepvo // occorrere  nella  Mercan- 
tia//insieme  con  vn  trattato // de'  Cambi.//  Et  in  somma  fi  ra- 
giona  di  //  tutto  quello  che  al  Mercante  Christiano  //  (1  conuiene. 
//  Composta  per  il  Dottor  Sarava,//  &  nuoamente  tradotta  di  lin- 
gua spagnuola  //  dal  S.  Alfonso  D' Vlloa.//  Cvm  Privilegio.//  In 
Venetia // Apreffo  Bolognino  Zaltieri.//  M  D  LXL'    (F.  i,  r.) 

Colophon.  *  In  Venetia  apreffo  Bolognino  Zaltieri // 1 561.' 
(F.  135,  V.) 

Description.  8°,  10. 2X  15. i  cm.,  the  text  being  7.6  x  12.2 
cm.     135  ff.  (3  unnumb.),  29  11.    Venice,  1561. 

Editions.    See  p.  229. 

This  is  an  Italian  translation  of  the  Spanish  edition  of  1544  (p.  229). 
The  book  was  evidently  written  in  1542  as  already  stated,  for  the  author 
speaks  (f.  109)  '  del  capo  quefto  anno  M  D  XLll.' 

MICHAEL   STIFEL.     Ed.  pr.  1545.  Nurnberg,  1545. 

See  p.  223. 

Title.    See  Fig.  112. 

ColopJion.  '  Zu  Nurnberg  Truckts  Johan  Petreius.//  1545.' 
(F.  96,  r.) 

Description.  4°,  15.5  X  20.7  cm.,  the  te.xt  being  10.8  x  15.5 
cm.     4  ff.  unnumb.  +  92  numb.  =  96  ff.,  32-34  11.     Nurnberg, 

1545- 

Editions.    See  p.  226. 

The  '  Deutsche  Arithmetica '  is  divided  into  three  parts.  In  the 
'  Haufsrechnung '  there  are  1 2  chapters,  all  relating  to  the  arithmetic 
of  the  common  j:)eople,  there  being  '  no  household  so  narrow  and  poor 
that  common  arithmetic  is  not  both  serviceable  and  necessary  to  its 
welfare.'    ('Es  ift  kein  Haufzhaltung/ niendart  fo  goring  vnnd  fchlecht/ 


232  RARA   ARITHMETICA 


rccljminci  t)cr  gan^cn  2(ritbmcticf )  bi§§cr  ^cn  '©cutfcfKn/mit  vit 
fi-cmbDcrt  ttjojtcn/vcrmcngt  vnD  vcrblcnD/fcf^nxr  tpgctt)crm/(5o 
trirt  fu  ^le  mit  nctt>  crfunt)ncm  vonbat  vnnD  D^cgcln  /  fc^:  fcicljf 
»nD  furl?  ^crfur  biac^f  vnD  ^dc  ^  it/  t)n^  mti  ^utcn  ^f  uf  fcf?cn  U» 
f antlicfjcn  tt)OJtm  t)n  ^7(nipc(n  crtucpfa.  ^c^e  anDcr  fo  fticrm 
^clcrt  win  von  Der -O^u^rcc^nwn^  i>nt5  ^ircfjcnrccfjuung/  fortgt 
fcincti  bcnc^f  ^nugfam  mit  ficf).Zlk$  tmcf)  -^m  S?icfea(l  (^ti* 
ft(/auff  cm  NfgnPac  ncn?c  vnD  Ccidjfc  tvct^  9c(^(l(cf» 

gu  t^iirn^er^  true?  f  $  3o§art  55«rcm^o 
CumPriufleg^o  ad  Qttlnquennium^* 

Fig.  112.   Title  pagk  of  stifel's  iJcutfchc  Arithmctica 


PRIjNTEI)    1U)()KS  233 

(las  jr  (lie  llaulzrechnung  odcr  gemeine  rechnung/  iiirhl  niit/.li(  h  vnnd 
dieftlich  sey.')  Stifel  first  treats  '\'om  Algorithmo  der  Recheiipfenning,' 
recommending  the  operations  as  being  '  wunderleichtlich  (lurch  die 
Rechenpfenning  gelernet  vnd  gelehret.'  The  coss  (algebra)  relates 
largely  to  arithmetic,  although  touching  upon  the  most  common  alge- 
braic operations  (Fig.  1 13).  In  it  Stifel  gives  the  rule  for  dividing  one 
fraction  by  another  by  using  the  inverted  divisor  as  a  multiplier.  The 
third  part  relates  to  the  ecclesiastical  calendar  ('Yonder  Kirchenrech- 
nung/die  man  nennet  Computum  Ecclefiafticum'),  a  subject  in  which 
he  acknowledges  his  indebtedness  to  '  Johannes  de  .Sacro  bufto.'  With 
his  usual  commendable  but  effusive  piety  Stifel  closes  with  praise  not 
only  to  the  Prince  but  to  '  vnferm  Vater  im  Hymel  vnd  feinem  eynigen 
naturlichen  Son/  vnferm  Herren  lefu  Chrifto.' 

JOHANN    SCHEUBEL.     Ed.  pr.  1545.  Leipzig,  1545. 

ScHEVRL,  ScHEUBKLius.  Bom  at  Kirchheim,  WUrttemberg,  August  iS, 
1494;  died  February  20,  1570.  Professor  of  Mathematics  in  the  University 
of  Tubingen,  to  which  institution  he  bequeathed  most  of  his  manuscripts. 
He  wrote  on  arithmetic  and  algebra,  and  edited  part  of  Euclid. 

Title.    See  Fig.  1 14. 

Colophon.  '  Lipfiae  ex  Officina  Michaelis // Blum,  a  reflittita 
fakite.// Anno  M.  D.  XLV.// Idib:  Maij.'    (F.  255,  r.) 

Dcscriptioti.  8°,  10  X  15.9  cm.,  the  text  being  6.9  x  12  cm. 
255  ff.  unnumb.  +  i   blank  =  256  ff.,  20-26  11.     Leipzig,   1545. 

Editions.  It  is  sometimes  stated  that  this  work  appeared  at 
Strasburg  in  1540,  but  I  cannot  verify  the  statement.  I  know 
of  no  edition  other  than  this  of  1545.  Scheubel  also  wrote  a 
'Compendium  arithmeticae '  (p.  246,  1549).  Murhard  mentions 
an  '  Arithmetica  sive  de  Arte  supputandi  Liber,'  Lipsiae,  1545, 
8^  but  ho  may  refer  to  the  '  De  Numeris.'  Scheubel  also  pub- 
lished an  algebra  (Paris,  155 1),  and  the  seventh,  eighth,  and 
ninth  books  of  Euclid  (1558). 

This  work  is  the  production  of  a  scholar  rather  than  a  man  con- 
versant with  the  demands  of  business.  While  Scheubel  tried  to  write 
a  mercantile  arithmetic,  the  result  was  far  removed  from  the  needs  of 
the  common  people.  It  carries  the  work  in  subjects  like  the  roots  so 
far  that  the  ordinary  Rechenmeister  could  not  have  used  it.  Moreover, 
it  is  written  in  Latin  and  is  much  more  extended  than  the  work  of 
Gemma  Frisius,  so  that  it  appealed  neither  to  the  business  school  nor 


2  34 


RARA   ARITHMETICA 


-H  vn^  -r  •  VIL 

4!)  uk  von  ^epc^cn  rcDctt  tDcrbe/foItu  mtc^  t)Cf(!(5n 
t)on  t)ifcrt  ieic^m  -f-  vnt> — /^cn  fomc^crKic^ 
me/  ettm:oDcr@um:XoDerft.:c'.  ^crDcicfe 
mc!jt5fr)d^ennenncn/fonDcrn/namm/ot)cr  bcnrtv 
nung  ^cr  ia(cn.5a}a  icij  nu  rct>c  von  glcic^en  icicle/ 
foltu  C0  vcrjicftn  von  -f-  vni)  -f-  /  oDor  von  —  vnD  — .  2((fo 
(juc^/ma  tcfe  von  vngWcfem  icjcftmf cDc  /  fo  vcrpcje  ce/von  -i- 
unb — . 

00  6<it>cn  nu  t)iTe  itvep  Sctcljen  -hvnD  —  /  tin  fontttUcf^m 
'^Ic^oiithmum/mkWf^  W  P^K^«  triKauff  4  OCegcfttv  5)cnn  er 
geHiet  lum  Qilgout^mo  Dcr  vngcrec^ncten  ^a(cn  tvi^Du  moU  fc^ 
fynmY^/mt>am$wa$  voiBingefagtip  vonDiTennamcn  fuin: 
fum:X  jf .  t)a$  ^t^out  aKw  §ic  5cr  /  ai$  vntcr  an  <mi3cn  Zi^o^ 
rifgmum. 

^«  erji  S^egcl  t^on  tern  2(bbtrm 

vnu  0uttra§irm.         VIII. 

big  yiid^m/  im  Tlbbmn  vn  ©ubtral)ircti/ 
o(>it  ^llem  (<>  bu  im  fubtral)it:ctt  btc  sal  /  bie 
b»  folrc/i  fttbttabitrc/iticljt  Eanfl  fiibtral^irc. 

^]ctnvpia  vom  2(t)t)iV€n. 


8  0um:  -h  7. 


0um: 


18. 


©urn:  —    18. 


H  @um: 


14. 


^K  fi^cj?  nu  vol  mm/m  -f-  vnt>  i-  macfec  itn  rrpcn  cr^ 

cinpta 

Fio.  113.    From  stifel's  Detitfchc  Arithmetica 


PRINTICl)    BOOKS  235 

to  the  ordinary  classical  school.  A  great  deal  of  attention  is  given  to 
exchange,  the  rule  of  three,  and  the  extracting  of  roots  of  high  order. 
Attention  is  also  given  to  problems  which  would  now  form  part  of 
algebra,  and  there  is  a  little  treatment  of  geometry  from  the  standpoint 
of  mensuration.    The  nature  of  the  work  can  be  somewhat  understood 

DENVME/ 

RIS    ET    DIVERSrS    RATIONIBVS 

feu  rcgulis  computationum  opufculum, 

a  loanne  Scheubelio  compofitum. 

Non  folum  ad  ufum  qucndam  uul 

garem/ed  etiam  cognitionem 

S^fcicntiam  exquuitiorem 

arithmetical  acco* 

modatum  • 


M.  D.  XLV. 

Fig.  114.    TrrLF.  i'ac.k  ok  sc^iikubki. 

from  the  titles  of  the  five  '  tractati '  :  1 .  '  De  numeris  integris  '  ;  2.  '  De 
proportionibus,  proportionalitatibus  (I^:  alijs '  ;  3.  '  De  minutijs  uulgari- 
bus  ' ;  4.  '  De  minutijs  phyficalibus  '  ;   5.  '  Aliquot  regulse.' 

While  Scheubel  is  not  much  appreciated  to-day,  he  was  really  ahead 
of  his  time.  He  tried  to  banish  the  expression  *  rule  of  three  '  and  to 
substitute  '  rule  of  projjortion.'    His  ex])Ianation  of  scpiare  root  is  in 


236  RARA   ARITHMETICA 

some  respects  the  best  of  the  century,  and  he  dismisses  with  mere  men- 
tion the  '  duplatio  '  and  '  mediatio  '  of  his  contemporaries.  He  extracts 
various  roots  as  far  as  the  24th,  finding  the  binomial  coefficients  by  means 
of  the  Pascal  triangle  a  century  before  Pascal  made  the  device  famous. 

SACROBOSCO.     Ed.  pr.  1545.  Wittenberg,  1550. 

See  p-  31- 

Title.  '  loannis  //  de  Sacrobvsto  //  Libellus  de  Sphaera.// 
Accessit  eivsdem  //  avctoris  Compvtvs  //  Ecclefiafticus,  Et  alia 
qusedam  //  in  ftudioforum  gra-//tiam  edita.//  Cum  Praefatione 
Philippi  //  Melanthonis.'  On  f.  6^ ,  r.,  begins  '  Libellvs  //  loannis 
de  Sacro//bvsto,de  Anni  Ratione,//sev  vt  vocatvr  vvl-//go  Com- 
pvtvs Ec-//clesiasti-//cvs.//Cvm  Praefatione // Philippi  Melan- 
//thonis.//Anno  M.  D.  XLV.' 

Colophon.  '  Impreffum  Vuitebergae  apud  //  lohannem  Crato- 
nem.//Anno//M.  D.  L.'    (F.  134,  r.) 

Description.  8°,  10.  i  X  15.5  cm.,  the  text  being  6.5  X  10.8 
cm.  134  ff.  unnumb.  +  2  blank  (2  plans)  =  1 36  ff.,  26  11.  Witten- 
berg, 1550. 

Editions.  The  preface  by  Melanchthon  is  dated  'Menfe  Auguf- 
to.//  Anno  //  M.  D.  XXXVIII,'  so  that  his  edition  of  the  book 
could  not  have  appeared  earlier  than  that  year.  The  fact  that  this 
volume  was  printed  in  1550,  as  shown  by  the  colophon,  makes  it 
probable  that  the  date  1 545  on  the  title  page  is  that  of  the  first  edi- 
tion. There  are  several  editions  of  the  'Sphaera'  in  Mr.  Plimpton's 
library,  but  this  is  the  only  one  containing  the  Computus. 

This  is  less  properly  included  in  a  list  of  arithmetics  than  many  of 
the  other  computi.  Since  several  others  have  been  included,  this,  which 
is  one  of  the  most  celebrated,  is  given  place  ;  but  the  arithmetical  work 
is  practically  nil. 

EUCLID.     Ed.  pr.  (arith.  books)  1545-  Rome,  I545- 

See  p.  II. 

Title.    See  Fig.  115. 

ColopJion.  '  Stampata  in  Roma  per  Antonio  Blado  Afolano.// 
M  D  XLV.'    (P.  12.) 


PRINTICI)    1U)()KS 


237 


Description.  8°,  9.8  X  15.1  cm.,  llie  text  bcinj;  6.7  X  10. 9  cm. 
112  pp.  (3  blank,  6  unnumb.),  24-26  11.     Rome,  1545. 

Editions.  There  were  several  editions  of  one  or  more  of  those 
books  of  Euclid  that  relate  to  some  part  of  the  theory  of  arith- 

I  QVINDICI 

LIBRI  DEGLI  ELEMEN 

TI     DI   EVCLIDE,    DI    GRE 

CO    TRADOTTI   IN 

tINGVA     THO- 

SCANA* 


IN  ROMA*  M  DXXXXV, 

Qonimu^i^rmXe^oiiti  S.  N.  S.  Pj^w/oTcr^, 

Ks  delta  Seretiisfima  republica  Venetiana 

^er  cinque  anni* 

Fk;.  115.    Title  p.xge  ok  tmk   1545   faclid 

metic.  Of  these  separate  books,  this  is  the  rare  first  Italian 
edition.  Among  others  may  be  mentioned  those  of  Wittenberg, 
1546;  1549;  Paris,  1551-4°  (P-  238);  ib.,  1554.  4°  (P-  238); 
1555  (by  Scheubel)  ;  Paris,  1557,  8°  (p.  240)  ;  Wittenberg.  1564, 


238  KARA   ARITHMETICA 

8°  (p.  240).  There  were  very  many  editions  of  Euclid's  work 
published  before  1600,  practically  all  including  Book  V,  on  pro- 
portion, and  some  including  the  other  arithmetical  books. 

Euclid's  treatment  of  arithmetic  was  purely  theoretical,  no  work  on 
computation  being  included.  This  copy  has  the  following  note  by 
De  Morgan,  who  once  owned  it :  "  This  book  was  also  printed  in  Greek 
—  same  year,  place,  size,  printer,  and  dedication.  There  is  a  Greek 
copy  in  the  British  Museum.    A.  De  Morgan,  Feb.  29,  1852." 

Other  works  0/1545.  Bseda,  p.  131,  1521  ;  Feliciano,  p.  149,  1526  ; 
Psellus,  p.  168,  1532;  Stifel,  p.  226,  1544;  Sfortunati,  p.  177,  1534; 
WiUichius,  p.  197,  1540;  Antonius  de  Barres,  '  Arithmeticae  practicae 
libri  IV,'  Louvain,  4°;  Johann  Obers,  '  Newgestelt  Rechenpiichlin,' 
Augsburg ;  Pedro  Espinosa,  '  Tractatus  proportionum,'  Salamanca,  fol. 

EUCLID.     Ed.  pr.  (arith.  books)  1545.  Paris,  1551. 

See  p.  II. 

Title.  See  Fig.  1 16.  (Note  the  signature  of  Giuliano  de  Medici.) 

Description.  4°,  16.3  X  23.4  cm.,  the  text  being  9.5  X  17.3  cm. 
162  ff.  (2  blank,  20  unnumb.),  29  11.    Paris,  155  i. 

Editions.    See  p.  237. 

This  is  not  the  same  as  the  1545  edition  already  described,  but,  like 
it,  this  relates  to  the  numerical  side  of  mathematics. 

EUCLID.    Ed.  pr.  (arith.  books)  1545.  Paris,  1554. 

See  p.  II. 

Title.  '  Evclidis  //  Elementa  qvaedam  //  Arithmetica.//  Lvte- 
ti£e,//Apud  Vafcofanum,  uia  lacobaea,  ad  infigne  Fontis.//  M.  D. 
LIIII.//Cvm  privilegio  regis.'  (F.  i,  r.) 

Description.  4°,  15.4X  19.4  cm.,  the  text  being  11.2  X  17.8 
cm.     18  ff.  numb.,  29  11.     Paris,  1554 

Editions.    See  p.  237. 

This  work  is  made  up  of  certain  extracts  from  the  various  books  of 
Euclid  relating  to  Arithmetic.  It  is  in  Greek  with  a  Latin  translation 
following  each  definition  or  theorem.  It  consists  of  such  standard  old 
definitions  as  '  Numerus  autem,  ex  unitatibus  composita  multitudo,'  and 
such  theorems  as  '  Omnis  primus  numerus,  ad  omnem  numerum  quern 
non  metitur,  primus  est.'  There  are  no  discussions,  illustrations,  or 
proofs  of  the  propositions. 


PRINTED    BOOKS  239 

Euclidiselemencorum 

LIBER    DECIMVS,   PET  RO 
Momaureo  intcrpretc. 

Ad  loannem  Bellaium  Cardinalcm. 


L  VTETIAE, 

^£ud  WdfcoftnumiUia  lacob^a  ad  injtgne  Tontif* 

M.     D.     LI. 
CVM    PRIVILEGIO, 

Fig.  116.   Title  page  of  the   155  i    euclid 


240  RARA   ARITHMETICA 

EUCLID.     Ed.  pr.  (arith.  books)  1545.  Paris,  1557. 

See  p.  1 1. 

Title.  '  Evclidis  //  Elementorvm  //  Libri  XV.  Grae-//ce  & 
Latine,//Quibus,  cum  ad  omnem  Mathematicae  fcientiae // par- 
tem, tum  ad  quamlibet  Geometriae  tra-//ctationem,  facilis  com- 
paratur  aditus.// E7rt7/9a/.t/xa  iraXaiov.l j  ^'yr)[x.aTa  irevre  \\\drw- 
vo<;,  a  IIvday6pa<i  ao-//<j)6<i  evpe.//  Uvda^opa'i  ao(^o<i  evpe,  HXdroJu 
S'  dpiSrjX'  i8i-//8a^ev  //  EuKXeiSi]^;  iirl  rolai  K\eo<;  7r€piKaX\e<i 
erev^ev.  Lvtetiae,// Apud  Gulielmum  Cauellat,  in  pingui  Gallina, 
//ex  aduerfo  collegij  Cameracenfis.//  1557.'    (F.  i,  r.) 

Description.  8°,  10.5  X  16.9  cm.,  the  text  being  7  X  12.8  cm. 
16  ff.  unnumb.  +  130  numb.  =  146  ff.,  25  11.     Paris,  1557. 

See  p.  238. 
EUCLID.     Ed.  pr.  (arith.  books)  1545.  Wittenberg,  1564. 

See  p.  1 1. 

Title.  '  Arithmetices  //  Evclideae  //  Liber  Primvs.//  Alias  in 
ordine  reliquorum // Septimvs  :  Qui  citra //prsecedentium  Sex 
librorum  //  Geometricorum  opem  erudite// perfequitur,  cum  reli- 
quis  duobus  //  fequentibus,  j.iera  principia  ac  //  folidiora  funda- 
menta  Logi-//ftices,id  eft,ut  uocant, // Arithmetices  Pra-//ctic3e. 
//Per //loan.  Sthen.  Luneb.//In  scholarvm  vsvm  Kara  to  on 
tractatus  ipcorrjfiartKO)^,  difquifitione  nimi-//rum  Dialectica  quae 
Dialogorum //est  propria.// 1564.'     (F.   i,  r.) 

ColopJwn.    'VVittenbergae.// Anno// 1564.'    (F.   106,  v.) 

Description.  8°,  9.2  x  14.8  cm.,  the  text  being  6.3  X  1 1.5  cm. 
107  ff.  unnumb.,  22-24  11.    Wittenberg,  1564. 

Editions.    See  p.  237. 

GASPARD  DE  TEXEDA.     Ed.  pr.  1546.    ValladoUd,  1546. 

A  ValladoUd  arithmetician  of  the  first  half  of  the  sixteenth  century. 

Title.    See  Fig.  117. 

ColopJion.  '  CFue  impreffala  prefente  //  obra  d'Arithmetica 
En  la  muy  noble //y  felice  villa  de  Valladolid  (Pincia // otro 
tiempollamada)  En  la  ofifici-//na  de  F'rancifco  Fernandez //de 


PRINTIU)    IK)()KS 


241 


on  i:jvmfc^m 


Fic.  117.   Title  page  of  texeda 


242  RARA   ARITHMETICA 

cordoua/ junto  alas // efcuelas  ma//yores // Acabofe  a  quatro 
dias  del  mes  //  de  Henero  defte  ano  del  //  fenor  de  mill  :  quini- 
//entos  2  quaren//ta  2  feys//Anos.'     (F.  Ixiiij,  r.) 

Description.  8°,  13.5  X  19.5  cm.,  the  text  being  10.3  x  16.5 
cm.  64  ff.,  numb,  in  Roman,  32-34  11.  Title  page  engraved  on 
wood.    Valladolid,  1546. 

Editions.  There  was  no  other  edition.  De  Morgan  (p.  103) 
gives  the  date  as  1545,  which  is  that  of  the  privilege,  the  colo- 
phon of  his  copy  having  been  torn  out. 

This  rare  Spanish  arithmetic  gives  the  fundamental  processes  with 
integers,  fractions,  and  denominate  numbers,  introduces  some  practical 
mensuration  under  the  title  '  De  Geometria,'  and  gives  a  rather  extended 
treatment  of  the  business  rules.  It  is  interesting  because  of  the  treat- 
ment of  Spanish  and  Arab  (algoristic)  notation,  or,  as  the  author  says, 
'de  numerar  en  caftellano  y  en  guarifmo.'  For  example,  his  two  methods 
of  writing  160,462,009,621,  are  : 

c.  Ix.  U462  qs.  .  ix  U621 

160  U462  qs  009  U  621.     (F.  iiij,  r.) 

PIETRO   CATANEO.     Ed.  pr.  1546.  Venice,  1546. 

A  sixteenth-century  arithmetician,  of  Siena. 

Title.    See  Fig.  118. 

Colophon.  'Stampato  in  Venetia  per  Niccolo  Bafcarini.// M  D 
XLVI.'     (F.  64,  r.) 

Description.  4°,  15.3  X  20.8  cm.,  the  text  being  12  x  16  cm. 
I  f.  unnumb.  -{-63  numb.  =64  ff.,  32-35  11.    Venice,  1546. 

Editions.  A  very  rare  edition,  and  probably  the  first  in  spite 
of  the  words  '  nvovamente  stampate,'  for  the  dedication  is  dated 
M.  D.  XLVI.  (F.  I,  V.)  There  were  two  later  editions,  Venice, 
1559.  4°  {P-  244),  and  Venice,  1567,  4°  (p.  244). 

The  work  is  fairly  practical,  and  in  many  respects  is  in  advance  of  its 
time.  Unlike  most  Venetian  books  it  uses  the  Florentine  name  '  biri- 
cvocolo  '  for  the  common  form  of  multiplication,  and  gives  the  'a  danda  ' 
division  before  the  galley  form,  recommending  it  as  '  molto  neceffario.' 
The  applications,  while  not  numerous,  are  practical,  and  throw  some 
light  upon  the  business  customs  of  Siena  and  Venice.  Cataneo  was  not, 
however,  an  original  writer.    His  arithmetic  is  composed  quite  largely 


PRINTED    HOOKS  243 

of  didactic  statements  to  be  found  in  the  works  of  his  predecessors,  and 
the  fact  that  he  gives  four  methods  of  multiplication  shows  that  he  could 
not  escape  the  influence  of  writers  like  Paciuolo. 

LE  PRATICHE  DELLE  DVB 
PRIME 
M  ATHEMATICHE 
Dl    PI  ET  RODE  CATANI 

DA  SIENA 


a 


LIBRO  D'ALBACO 
E  GEOMETRIA 


NVOVAMENTE    STAMPATE, 


INVENETIA  M     P     y.LVl 

Fig.  118.   Title  page  of  the   1546  cataneo 

Other  works  of  1546.  Anonymous,  p.  213,  1542;  Boethius,  p.  27, 
1488;  Euclid,  p.  237,  1545;  Glareanus,  p.  192,  1539;  Helmreich, 
p.    303,    1561  ;    Manzoni,    p.   257,    1553;    Misrachi,    p.    180,    1534; 


244  RARA   ARITHMETICA 

Rudolff,  p.  152,  1526;  Sole,  p.  143,  1526;  Stifel,  p.  226,  1544;  Anony- 
mous, '  An  introduction  for  to  lerne  to  recken  with  the  pen,  or  with  the 
counters  accordyng  to  the  trewe  cast  of  Algorisme,  in  hole  numbers,  or 
in  broken,  newly  corrected.  And  certayne  notable  and  goodly  rules  of 
false  positions  thereunto  added,  not  before  sane  in  our  Englysche  Tonge,' 
I>ondon,  8°,  with  another  edition  at  London  in  1574,  sm.  8°;  Anony- 
mous, '  Ein  new  kurtz  Rechenbtichlein  auff  der  Linien  und  Federn,' 
Frankfort,  8°,  possibly  by  Giilfferich  (p.  269,  1555);  Alfonzo  Lopez  de 
Corella,  '  Secretos  de  las  cuatro  mathematicas  ciencias,'  Valladolid. 

Works  of  I ^4"/.  Gemma,  p.  200,  1540;  Saravia,  p.  229,  1544; 
Tagliente,  p.  114,  15 15. 

Works  0/1348,  Gemma,  p.  200,  1540;  Ghaligai,  p.  132,  1521; 
Riese,  p.  139,  1522;  Stifel,  p.  226,  1544;  Tagliente,  p.  114,  1515; 
Uberti,  see  Tagliente,  p.  114,  1515  ;  Wolphius,  p.  154,  1527;  Anony- 
mous, '  Specie  principali,  et  primi  principii  del'  Arithmetica  di  C.  de 
C.  P.,'  Bologna. 

PIETRO  CATANEO.     Ed.  pr.  1546.  Venice,  1559. 

See  p.  242. 

Title.  '  Le//  pratiche  //  delle  dve  prime  //  Matematiche  //  di 
Pietro  Cataneo  //  con  la  aggionta,//  libro  d'Albaco  e  Geome- 
tria  con  il//pratico  e  uero  modo  di  mifurar  la  Terra.// Non  piv 
mostra  da  altri.//  (Woodctit  of  grififin  with  motto  :  '  Virtute  dvce 
//comite  Fortvna.')  In  Venetia,  apreffo  Giouanni  Griffio,  MD 
LIX.'     (F.   I,  r.) 

Colophon.  '  In  Venetia,  apreffo  Giouan  GrifBo,  ad  inftantia// 
di  M.  Pietro  Cataneo,  M  D  LIX.'   (F.  83,  v.) 

Description.  4°,  15.3  X  20.3  cm.,  the  text  being  11.4X  15.1 
cm.  I  f.  unnumb.  +  82  numb,  -f  i  blank  =  84  IT.,  32-35  11. 
Venice,  1559. 

See  p.  242. 

PIETRO  CATANEO.     Ed.  pr.  1546.  Venice,  1567. 

See  p.  242. 
Title.    '  Le  //  pratiche  //  delle  dve  prime //Matematiche  //  di 
Pietro  Cataneo  Senese,//  ricorrette,  &  meglio  ordinate,  con  alcu- 
ne  ag-//giontioni  de  lo  steffo  Autore.//Diuife  in  libri  quattro.// 


PRINTED    HOOKS  245 

(Woodcut  of  <;rirrin,  with  nioUo  :  '  Virlvlc  (lvc-c,//coniilc  For- 
tvna.)  In  Vonetia,  aprcffo  Giouanni  (}ritTio,//M  D  LW'II.' 
(F.    I,   r.) 

Colophon.  '  In  Venetia,  apreffo  Giouan  Griffo,  ad  inftanlia  di 
//M.  Pictro  Cataneo,  M  D  LXVIL'    (F.  88,  r.) 

Description.  4°,  15.5  X  20.7  cm.,  the  text  being  11.7  x  15.7 
cm.    88  ff.  numb.,  32-35  11.    Venice,  1567. 

Editions.  See  p.  242.  This  differs  from  the  first  (1546)  edi- 
tion only  in  the  fact  that  the  part  relating  to  geometry  contains 
considerable  additional  matter. 

See  p.  242. 

JACQUES   PELETIER.     Ed.  pr.  1549.  S.  1.,  1607. 

Bom  at  Mans  in  1517  ;  died  at  Paris  in  July,  1582.  Tie  became  principal 
of  a  college,  traveled  extensively,  and  contributed  both  to  literature  and 
to  elementary  mathematics. 

Title.  '  L'Arithmeti-//qve  de  lacqves  //  Peletier  dv  //  Mans, 
//Departie  en  quatre  liures.Z/Troifieme  edition,  reucue  et  aug- 
mentee.//Par  lean  de  Tovrnes.// M.  DC.  VII.'    (P.  3.) 

Description.  8°,  10.3  X  16  cm.,  the  te.xt  being  T .7  ^  13  cm. 
297  pp.  (6  blank,  i  unnumb.),  26  11.     S.  1.,  1607. 

Editions.  Poitiers,  1549,  4°  (Graesse,  Sup.);  ib.,  1552,  8° 
(Graesse  says  155  i);  Lyons,  1554,  8°.  Graesse  mentions  all  of 
these,  but  the  above  title  shows  this  edition  of  1607  to  be  the 
third,  possibly  the  third  revision.  I  have  also  seen  mentioned 
an  edition  of  1567,  4°,  a  Lyons  edition  of  1570,  and  a  Latin  edi- 
tion at  Paris  in  1563  and  1578. 

The  work  is  quite  practical,  although  it  contains  a  considerable 
amount  of  medi3eval  matter.  The  first  book  treats  of  the  fundamental 
operations  with  integers,  the  second  of  fractions,  the  third  of  roots  and 
proportion,  and  the  fourth  of  the  applications  of  arithmetic.  It  contains 
a  number  of  such  traditional  problems  as  the  hare  and  hound.  Peletier 
also  wrote  a  chapter  '  De  fractionibus  astronomicis  compendium  de 
cognoscendis  per  memoriam  calendis,'  that  was  published  in  his  editions 
of  Gemma  Frisius.  Graesse  mentions  an  '  Arithmeticae  modus,'  Paris, 
1563,  8°,  probably  the  Paris  edition  referred  to  above.  Peletier  also 
wrote  one  of  the  first  practical  textbooks  on  algebra. 


246  RARA   ARITHMETICA 

JOHANN  SCHEUBEL.     Ed.  pr.  1549.  Basel,  1549. 

See  p.  233. 

Title.  The  title  page  is  missing  in  this  copy,  but  the  running 
headline  is  '  Compendium  Arithmeticae.'   (See  the  1560  edition.) 

Colophon.  '  Basileae,  per  lacobvm  //  Parcvm,  expensis  //  loan- 
nis  Oporini,// Anno  I549-'     (F-  87,  r.) 

Description.  Sm.  8°,  8.7  X  13.7  cm.,  the  text  being  6.3  X  1 1.8 
cm.     87  ff.  unnumb.,  27  11.    Basel,  1549. 

Editions.  Basel,  1549,  sm.  8°  (here  described);  ib.,  1560,  8° 
(below).  That  this  is  the  first  edition  is  seen  in  the  '  Epiftola 
Dedicatoria,'  which  bears  date  '  Tubingae,  idibus  Marti]  //  annni 
fefqui  millefimi // quadragefimi  noni.'     (P.  7.) 

While  open  to  some  of  the  criticism  mentioned  in  connection  with 
Scheubel's  '  De  mmieris '  (  p.  233,  1545),  this  book  is  more  practical 
than  his  earlier  one,  and  was  enough  in  demand  to  warrant  two  editions. 
It  is  not,  however,  a  commercial  textbook. 

JOHANN   SCHEUBEL.     Ed.  pr.  1549.  Basel,  1560. 

See  p.  233. 

Title.  '  Compen-//divm  Arithme-//ticae  Artis,  vt  bre-//uif- 
sum  ita  longe  utilifsimum  eru//diendis  tyronibus,  non  folum  pro- 
//pter  ordinem,  quo  paucis  perftrin-//guntur  omnia  huius  artis 
capita :  fed  //  etiam  caufa  perfpicuitatis,  quae  plu-//rimum  de- 
lectat  &  iuuat  difcentes,//  fummopere  expetedum  :  per  loan-// 
nem  Scheubeliu  adornatum//&  confcriptum.//Iam  denuo  ab  ipfo 
autore  recognitum  //  &  emandatum.//  Continent  autem  utrunq5 
hoc  Compendia,// numerorum  fcilicet  &  calculorum,  feu//pro- 
iectilium  (ut  uocant)  ra-//tiocincationem.//  Bafilae,  anno  1560.' 
(P.   I.) 

Colophon.  '  Basiliae  //  excudebat  lacobus  Parous,//  expenfis 
loannis  Opo-//rini,  anno  M.D.LX.//menfe  Martio.'    (P.  205.) 

Description.  8°,  9.5  X  15  cm.,  the  text  being  6.1  X  12  cm. 
14  pp.  unnumb.  +  191  numb.  (3-193)  +  i  blank  =  206  pp.,  23- 
24  11.    Basel,  1560. 

See  above. 


PRINTED    BOOKS  247 

JOHANN   FISCHER  (Piscalor). 

Ed.  pr.  1549.  Stettin,  s.  a.  (1565?). 

A  German  Rechenmeister  of  the  second  half  of  the  sixteenth  centui^. 

Title.  '  Ein  kurtz  Rechenbuchlein  fur  die  anfahendc  Schulcr 
gemacht  //Durch  Johann  Fifchcr.  Gcdruckt  zu  Alton  Stettin// 
in  Johan  Eichorns  Druckerey.'     (F.  i,  r.) 

Dcscnption.  8°,  9.8  x  15.4  cm.,  the  te.xt  being  6.^  x  1 1.2  cm. 
16  ff.  unnumb.,  Stettin,  s.  a.  (1565  ?). 

Editions.  This  work  appeared  first  in  Latin  under  the  title 
'  Arithmeticae  Compendium,  pro  Studiosis  hujus  artis  tyronibus 
recognitum/  Leipzig,  1549,  with  subsequent  Latin  editions,  ib., 
1554.  1559.  1582,  1592,  1598,  and  Wittenberg,  1592,  all  8°. 
Of  the  German  translation  there  were  editions  as  follows  :  Stettin, 
s.  a.  (1565  .?),  8°  (here  described) ;  Frankfort  an  der  Oder,  1566  ; 
Leipzig,  1 58 1,  8°;  1592,8°.  Fischer  also  published  a  work,  said 
to  be  different  from  the  '  Compendium,'  entitled  '  Ein  kiinstlich 
Rechenbijchlein,'  Wittenberg,  1559,  with  four  editions  from 
1559  to   1592. 

As  the  title  suggests,  this  is  merely  a  compendium,  designed  to  serve 
as  an  introduction  to  practical  arithmetic.  It  has  no  more  merit  than 
any  brief  primer. 

JUAN   DE  YCIAR,  Vizcayno. 

Ed.  pr.  1549.  Saragossa,  1549. 

IciAR.  A  Basque  arithmetician,  born  at  Durango  in  1525.  On  f.  3,  r., 
is  a  large  portrait  with  the  inscription  '  loannes  de  Yciar  astatis  sve  anno  xxv.' 
He  lived  in  Saragossa,  and  was  well  known  as  a  calligrapher. 

Title.    See  Fig.  119. 

Colophon.  '  CFue  impreffo  el  prefente  libro  en  la  muy  noble 
//y  leal  ciudad //d'^arago^a  en  cafe  de  Pedro  Bernuz/a  cofta 
//del  auctor  y  de  Miguel  de  ^apila  mercader  d'libros,// Acabo 
fe  a  .xvj.  de  Febrero  del  ano  de  mil  y  //quinientos  y  quarenta  y 
//nueue.'    (F.  61,  v.) 

Description.  Fol.,  19  X  28.9  cm.,  the  te.xt  being  14.3  X  21.5 
cm.  4  ff.  unnumb.  +  56  numb.  +  3  blank  =  63  ff.,  27-38  11. 
Saragossa,   1549. 


248 


KARA   ARITHMETICA 


Fig.  119.   Title  page  of  yciar 


PRiNrr:i)  hooks  249 

Editions.  Saragossa,  i  549,  fol.  (here  dcscribccl)  ;  ih.  1555,4°; 
ib.,  1564,  4°.  The  book  probably  had  several  other  editi(jns,  for 
Heredia  (I,  154)  says  that  this  is  'una  des  plus  anciennes  edi- 
tions de  ce  traite.' 

There  are  numerous  interesting  features  in  this  book.  Among* these 
is  Yciar's  fanciful  explanation  of  the  origin  of  the  Roman  numerals, 
part  of  it  traditional,  as  that  V  stands  for  five  because  it  was  the  fifth 
Latin  vowel,  and  part  more  recent,  as  that  L  was  half  of  the  old  form 
for  C.  U  is  used  instead  of  M,  as  with  several  Spanish  writers,  and 
cuento  is  used  for  million  as  was  their  general  custom.  (See  p.  60, 
Ciruelo.)  The  treatment  of  the  fundamental  operations  is  followed  by 
progressions,  compound  numbers,  roots,  mensuration,  and  such  common 
applications  as  '  las  compahias  fin  tiempo '  and  •  con  tiemjw,'  and  '  las 
reglas  de  teftamentos.' 

Other  works  of  IS49-  Anianus,  p.  32,  1488;  Anonymous,  p.  213, 
1542  ;  Boethius,  p.  27,  1488  ;  Euclid,  p.  237,  1545  ;  Gemma,  p.  202, 
1540;  Glareanus,  p.  192,  1539  ;  Kobel,  p.  102,  15  14  ;  Recorde,  p.  214, 
c.  1542  ;  Hans  Bock,  '  Ein  new  Rechenbiichlein  aufif  der  Linien  und 
Federn,'  Niirnberg  (probably  a  second  edition  of  the  1544  book,  p.  229)  ; 
Joannes  Stigelius,  '  Arithmetica,'  Leipzig,  8°,  with  a  second  edition,  s.  1., 
1554  (Victorinus  Strigelius  of  1563  ?  see  p.  311)  ;  Juan  Vejar,  'Arith- 
metica practica,'  Saragossa,  4°.    (Yciar's  work?) 

VALENTIN   MENIIER  de  Kempten. 

Ed.  pr.  1550.  Antwerp,  1565. 

Mennher.    a  German-Dutch  arithmetician  of  the  sixteenth  century.    See 
also  p.  281,  1556. 

Title.  '  Practicqve // pour  brievement //apprendre  a  Ciffrer, 
&  tenir  Liure  //  de  Comptes,  auec  la  Regie  de  //  Cofs,  &  Geome- 
tric.//Par  M.  V.  Menher  Alleman.// (Woodcut  of  counting 
house.)  A  Anvers,  I'an  M.  D.  LXV.//Auec  priuilege  du  Roy 
pour  4  ans.'  (E.  i,  r.)  Bound  with  this  in  the  third  part  is 
'La  Regie  d'Algebra,'  or  'Cofs.'  Also  the  '  IVacticqve // des 
Triangles // Spheriqves.// .  .  .  Anvers  .  .  .  M.  D.  LXIIIL' 

Colophon.  '  Imprime  en  Anuers  par  yEgidius  Dieft,//rAn 
de  noftre  Seigneur  lefu  Chrift.//M.  D.  LXV.// 19.  lanuarij.' 
(F.  113,  V.) 


250  RARA   ARITHMETICA 

Description.  8°,  9.6  X  i  5.6  cm.,  the  text  being  7.3  X  12.6  cm. 
113  ff.,  26  11.  The  algebra  contains  120  ff.  unnumb.  (part  3); 
the  geometry,  102  ff.  (part  4),  besides  the  '  Practique  des  tri- 
angles spheriques.'    Antwerp,  1565. 

Editions.    Menher  wrote  three  or  four  arithmetics,  as  follows  : 

1.  '  Practique  briesve  pour  cyfrer  et  tenir  Livres  de  compte,' 
Antwerp,  1550,8°;  1556  (probably  the  one  mentioned  on  p.  281); 
Antwerp,  1565,  8°  (here  described).  Unlike  the  De  Morgan  copy 
this  does  not  have  1564  for  1565  in  the  colophon.  Indeed  I 
think  De  Morgan  probably  looked  at  the  colophon  of  the  geome- 
try instead  of  the  arithmetic. 

2.  'Arithmetique  seconde,' Antwerp,  1556  (p.  281).  A  com- 
parison of  this  with  the  '  Practique '  shows  it  to  be  substantially 
the  same  work.  In  his  epistle  to  the  reader,  Menher  speaks 
of  the  1565  edition  of  the  'Practique'  as  merely  a  revision  of 
'  noftre  feconde  Arithmetique  de  I'an  M.  D.  LVL' 

3.  'Livred'Arithmetiqve,' Antwerp,  8°,  1573  (p.  347).  I  know 
of  no  other  sixteenth-century  edition  of  this  work,  although  there 
was  a  Rotterdam  edition  in  1609,  8°.  The  work  is  entirely  differ- 
ent from  the  '  Practique.' 

4.  '  Arithmetica  Practice,'  Autorff,  i  560,  8°.  I  know  nothing 
of  this  work. 

The  '  Practique  '  and  the  '  Arithmetiqve  seconde,'  essentially  the 
same  work,  are  mercantile  textbooks,  possessed  of  the  spirit  of  the 
« Livre  d'Arithmetiqve '  (p.  347 ),  but  not  as  successfully  written, 
Menher  was  one  of  the  pioneers  among  the  Dutch  arithmeticians,  and 
his  successors,  particularly  in  the  period  from  1600  to  1650,  produced 
some  very  practical  textbooks. 

ADAM  RIESE.     Ed.  pr.  1550.  Leipzig,  1550. 

See  p.  138. 

Title.    See  Fig.  120. 

Colophon.  '  Gedruckt  zu  Leipzig  durch //Jacobum  Berwalt.' 
(F.  196,  r.) 

Description.  4°,  15.5  X  17.8  cm.,  the  text  being  10.6  X  14.4 
cm.   4  ff.  unnumb. -f- 196  numb.  =  200  ff.,  29-31  11.    Leipzig,  1550. 


PRINTED    1U)()KS 


251 


tm3e/aufft>m£mt§m 

5Dariu  fotttH  t>n5  bc^mWgf  ejft>utc6  W< JOwpotftio* 
mW-PMcfica  genant/SWtt  grflntftcfetm 

C^urc^  2(bam  SKwfetu     c: 


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Cum  gratia  &  priuikgfo 
Cgffarco# 

Fig.  120.   Title  page  of  the   1550  kiese 


252  RARA   ARITHMETICA 

Editions.  See  p.  140.  This  is  the  fourth  of  Riese's  books  (p. 
139).  Bound  with  it  is  Isaac  Riese's  arithmetic  of  1580  (p.  365). 
The  date  is  also  given  in  the  dedicatory  epistle,  '  im  1550  jhar.' 

The  first  forty-six  folios  contain  the  treatise  '  auff  den  Linihen,'  the 
counter  reckoning.  This  is  followed  (ff.  47-105)  by  that  'auff  der 
Feder,'  the  common  algorism.  The  third  part  is  the  '  Practica,'  and 
the  fourth  the  '  Vifieren  '  or  gauging.  The  book  represents  the  cul- 
mination of  Riese's  work,  and  is  the  best  exponent  of  the  practical 
arithmetic  of  the  middle  of  the  century  in  Germany. 

Other  works  0/1550.  Agricola,  p.  171,  i533;  Agrippa,  p.  167, 
1531;  Anianus,  p.  32,  1488;  Borghi,  p.  22,  1484  ;  Cassiodorus,  p.  21 1, 
1540;  Feliciano,  p.  149,  1526;  Gemma,  p.  200,  1540;  Glareanus, 
p.  192,  1539;  Lonicerus,  p.  253,  1551;  Medlerus,  p.  223,  1543; 
Regius,  p.  181,  1536;  Riese,  p.  139,  1522;  Taghente,  p.  114,  15 15  ; 
Torrentini,  p.  76,  1501.  There  was  also  an  edition  of  Sfortunati,  c.  1550 
(p.  174,  1534).  Two  other  works  published  c.  1550  should  be  men- 
tioned :  Anonymous,  '  Opera  che  insegna  a  tener  conto  de  libro  secondo 
lo  cosueto  di  tutti  li  lochi  della  Italia  al  modo  mercantile,'  s.  1.  a.,  with 
some  mercantile  arithmetic;  William  Buckley,  'Arithmetica  memorativa 
sive  compendaria  Arithmeticae  tractatio,'  8°,  s.  1.  a.,  but  later  in  Seton's 
Logic,  London,  1572,  1574,  1577,  1584,  8°. 

JOHANN   SCHEUBEL.      Ed.  pr.  1551.  Paris,  1551. 

See  p.  223. 

Title.  '  Algebrae//compendiosa//facilisqve  descri-//ptio,  qua 

depromuntur  magna  //  Arithmetices  miracula.//  Authore  loanne 

Scheubelio  Mathematicarum //prof eff ore  in  academia  Tubingenfi. 

//Parisiis,//Apud  Gulielmum  Cauellat,  in  Pingui  Gallina,//ex 

'  aduerfoCollegiiCameracenfis.//i55i.//Cvmprivilegio."  (F.  i,r.) 

Colophoji.  '  Excudebat  Lutetiae  Parifiorum,  Benedictus  Preuo- 
tius  Typo-//graphus,invico  Frementello,  fub  infigni  ftelte  aureae. 
//1551.'    (F.  numb.  52,  v.) 

Description.  4°,  13.2  x  18.5  cm.,  the  text  being  9.4  X  15.2 
cm.     52  ff.  numb.,  32-37  11.    Paris,  1551. 

Editions.  There  was  no  other  edition. 

I  have  included  this  algebra  because  it  contains  some  work  in  the 
extracting  of  roots  by  the  galley  method,  and  therefore  shows  the  per- 
sistence of  this  mediaeval  plan. 


PRINTED    BOOKS  253 

ROBERT   RECORDE.     Ed.  pr.   1551.  London,  1596. 

See  p.  213. 

Title.  'The  Castle //of // Knowledge.// To  Knowledge  is 
this  Castle  fet,//  All  Learnings  friends  wil  it  fupport,//  So 
fhall  their  name  great  honour  get,//  And  gaine  great  fame  with 
good  report. //Though  fpitefull  Fortune  turn'd  her  wheele,// 
To  ftay  the  Sphere  of  Vranie,//Yet  doth  the  fphere  refift  that 
wheele,// And  flee'th  all  Fortunes  villanie // Though  earth  do 
honour  Fortunes  ball,// And  beetles  blinde  her  wheele  aduance, 
//The  heauens  to  Fortune  are  not  thrall,//  The  fpheres  furmount 
all  Fortunes  chance.// London // printed  by  Valentine  Sims, 
afsigned // by  Bonham  Norton.// 1596.'     (P.  i.) 

Colophon.  '  Imprinted  at  London  by  Valentine  //  Simnies. 
1596.'    (P.  236.) 

Description.  4°,  14.4  x  19.2  cm.,  the  text  being  12.5  x  17.5 
cm.  3  pp.  unnumb.  +  I  blank  +  232  numb.  =  236  pp.,  41  11. 
London,    1596. 

Editions.  London,  1551  ;  ib.,  1556,  fol.;  ib.,  1596,  4°  (here 
described). 

Recorde's  name  does  not  appear  on  the  title  page,  but  he  signs  the 
letter  of  dedication  to  '  Princesse  Marie,'  '  Robert  Record  Phyficion.' 
The  work  is  on  astronomy,  and  is  of  interest  in  the  history  of  arith- 
metic only  in  the  operations  involving  sexagesimal  fractions.  Division 
is  performed  by  the  galley  method,  and  there  are  no  symbols  for 
degrees,  minutes,  and  seconds.  The  arithmetical  part  includes  the  rule 
of  three.  Like  the  author's  other  works  (pp.  213,  286),  this  is  in  the 
catechism  form. 

Other  works  of  i^^i.  Borghi,  p.  i6,  1484  ;  Euclid,  j).  238,  1545  ; 
Gemma,  p.  200,  1540;  Glareanus,  p.  193,  1539  ;  Noviomagus,  p.  197, 
1539;  Peletier,  p.  245,  1549;  Recorde,  p.  214,  c.  1542;  Tonstall, 
p.  134,  1522;  Adam  Lonicerus,  '  Arithmeticae  brevis  Introductio,' 
Frankfort,  8°,  with  subsequent  editions,  ib.,  1568,  8°  ;  1570,  12°  ;  1581 ; 
1585  ;  1600,  8°  (Tropfke  puts  the  first  edition  as  1550)  ;  Andres  Garcia 
de  Lovas,  'Tratado  del  computo,'  Salamanca,  8°  ;  Innocenzo  Ringhieri, 
'  Centi  givochi  liberali  ...  in  dieci  libri  descritti,'  Bologna,  4°,  with 
subsequent  editions,  Venice,  1553,  4°;  Bologna,  1580;  Lyons  (French 
translation),  1555,  4°  (contains  some  number  games). 


254  RARA   ARITHMETICA 

MARCO  AUREL.     Ed.  pr.  1552.  Valencia,  1552. 

Aurel  was,  as  he  states,  a  German.    He  lived,  however,  for  several  years 
in  Valencia,  and  published  a  work  there  in  1541. 

Title.    See  Fig.  121. 

Description.    4°,  14.5  x  19.3  cm.,  the  text  being  9.9  X  16.6  cm. 
4  ff.  unnumb.  +  140  numb.  =  144  ff.,  31-36  11.    Valencia,  1552. 
Editions.    There  was  no  other  edition. 

Aurel,  in  his  letter  to  the  reader,  dwells  upon  the  unfortunate  state 
of  mathematics  in  Spain,  and  says  that  he  feels  called  upon  to  assist  in 
making  known  a  science  so  necessary  to  humanity.  Of  the  twenty-four 
chapters  in  the  book,  the  first  six  may  be  said  to  relate  to  arithmetic  as 
we  ordinarily  consider  it,  the  rest  referring  entirely  to  algebra.  The 
arithmetical  chapters  present  the  subject  in  a  fairly  practical  way,  but 
are  deficient  in  genuine  problems.  To  subtract  is  called  '  Restar,'  as  at 
present  in  Spanish,  the  same  root  appearing  occasionally  in  other  lan- 
guages, and  our  word  '  rest '  (for  remainder)  being  a  relic  of  this  name. 
Division  is  performed  entirely  by  the  galley  method,  but  the  figures  are 
not  canceled  as  is  generally  the  case.  '  Proporcion '  is  used  for  ratio, 
and  *  proporcionalidad '  for  proportion,  as  was  generally  the  custom  in 
the  early  arithmetics  of  all  Latin  countries,  a  custom  derived  from  the 
Boethian  books.  The  '  proporciones  '  (ratios)  are  treated  at  consider- 
able length  after  the  fashion  set  by  the  mediaeval  writers.  The  applica- 
tions are  almost  entirely  under  the  '  Regla  de  tres  '  (rule  of  three). 

In  the  part  devoted  to  algebra,  surd  numbers  are  first  treated,  the 
root  symbols  showing  the  German  influence.  The  plus  and  minus  signs 
are  also  used  as  extensively  as  in  the  works  of  writers  like  Stifel  and 
Scheubel,  and  the  symbols  for  the  various  powers  of  the  unknown 
quantity  are  such  as  are  found  in  the  works  of  contemporary  writers  in 
other  countries. 

LILIUS  GREGORIUS  GYRALDUS. 

Ed.  pr.  1552.  Venice,  1553. 

A  philosopher  of  Ferrara,  of  the  middle  of  the  sixteenth  century. 

Title.    See  Fig.  122. 

Description.  8°,  9.3  X  15.5  cm.,  the  te.xt  being  6.1  X  12.4  cm. 
184  pp.  (2  blank,  16  unnumb.),  29  II.    Venice,  1553. 

Editions.  Venice,  1552,  8°;  ib.,  1553,  8°  (here  described). 
This  treatise  also  appeared  in  his  'Opera,'  Basel,  1580,  sm.  fob, 
and  Leyden,  1696. 


?*'LIBRO    PR  I- 

ME^O,    VE     J<IilTHMETlCJ 

Mgehratica  ,€nel  qualfe  contlene  el  arte  Mercanituol, 

con  otras  muchas  ^glas  del  arte  menor^yla^egla  del 

jflgebra ,  Vulgarmentellamada  Arte  mayor  ^  o  \egla  de 

la  CO  fa  ifin  la  qual  no  fe  podra  entendet  el  dccimo  de  Eu-> 

elides ,  ni  otros  muchos  primores ,  a/Tt  en  jfrithmetica  co* 

mo  en  Geometria :  compuejlo  ,ordenado  ,y  hecho  Impri* 

mir  por  iVfarco  Aurel,  natural  Meman :  Intitulado,  Ve6 

jpertador  deingenios.  Vadiriffdo  al  muy  ma^i* 

fico  fmormoffen  Bernardo  Cimon,  Cm» 

Jadano  dela  mi{y  infigney  co* 

ronada  Gudad  df 

Valencia^ 


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^  Con  Priuilcgio  de  fu  Mageftad, 
por  ticmpo  dc  dlcz  anos« 


En  cafa  de  loan  de  Mcy,  Mandroi 


Fig, 


12  1.   Title  page  of  aukel 


256 


KARA   ARITHMETICA 


LILII     GREGORII 

GYRALDI    FERRARIEN.     SVAa 

R  V  M    Q.V  ARVNDAM     ANN  Ob 

tationum  Dialogifmi  XXX.  ad  Awa 
plij?.  C(xrd>  Saluiatum, 

Item  Uurentij  frizzolij  Solmenfls  Dialogifmui  unicttt 
dc  ipjlm  Ulij  uiu  cr  opmbm* 


V  E  N  E  T 1 1 5,  Apud  Gmltcrum  Scottm, 

M     D     L     I     I     I. 
^1 


Fig,  122.   Title  page  of  the   1553  gyraldus 


PRlNTI':i)    HOOKS  257 

This  set  of  dialogues  is  of  interest  in  the  history  of  mathematics  in 
that  the  second  and  third  parts  deal  with  notation  and  finger  symbolism. 
'  Dialogismvs  secvndvs  de  manus  &  digitorum  nominibus  d(§q;  nume- 
randi  per  eos  antiquorum  ratione  '  (p.  10)  is  an  almost  unknown  sketch  of 
finger  symbolism.  'Dialogismvs  tertivs  ad  Baptistam  Lucarinum  FR. 
filium  optimre  fpei  ac  indolis  puerum,  de  notis  &  finguris  numerorum, 
quibus  Latini  ac  Grseci  utebantur '  (p.  20)  is  an  eciually  interesting 
sketch  of  the  Greek  and  Latin  numerals.  Gyraldus  also  published  a 
♦Brevis  instructio  de  Grecor:^  numerali  supputatione '  in  15 13,  fol. 

Other  tvorks  0/1552.  Agricola,  p.  171,  i533  ;  Gemma,  p.  203, 
1540;  Ghaligai,  p.  132,  152 1  ;  Herman  GiillTerich,  '  Ein  new  kurtz 
Rechenbiichlein,'  Frankfort,  8°,  with  editions  ib.,  1555,  12°,  1568,  8° 
(see  also  p.  244,  1546,  and  p.  292, 1559);  Ortega,  p.  94,  1512  ;  Peletier, 
p.  245,  1549  ;  Recorde,  ]).  214,  c.  1542  ;  Riese,  p.  139,  1522  ;  Joachim 
Camerarius,  '  Arithmologia,'  i  2°  (there  was  also  an  edition  at  Basel,  s.  a., 
16°);  Dunkel,  '  Arithmetica,'  Leipzig,  8°;  Christopher  Falconius,  '  Re- 
chenbuch,'  Konigsberg  in  PreufJen,  4°  (Murhard  cites  as  a  different 
work  of  the  same  date  '  Rechenbuch  auff  die  Preusche  mijntz  mass  und 
Gewicht,'  ib.,  4°). 

DOMENICO  MANZONL     Ed.  pr.  1553.    S.  1.  (Venice),  i  553. 

A  sixteenth-century  arithmetician,  born  at  Oderzo. 

Title.    See  Fig.  123. 

Description.  8°,  10.5  X  15.8  cm.,  the  text  being  8.1  X  13.4  cm. 
16  ff.  immimb.,  26  11.  except  where  arranged  in  sections.  S.  1. 
(Venice),  1553. 

Editions.  There  was  no  other  edition  of  this  work.  Manzoni 
had  already  pubUshed  a  textbook  entitled  '  Libretto  molto  utile 
per  imparar  a  leggere,  scrivere  et  Abaco,  con  alcuni  Fondamenti 
della  Dottrina  Christiana,'  Venice,  1546,  8°,  31  pp.  of  which  are 
devoted  to  arithmetic.  He  also  published  in  1553  a  more  elabo- 
rate treatise,  '  La  Brieve  Risolvtione  di  Aritmetica  universale 
in  qualsi  voglia  negotio,  doue  intcrucnga  numero,  peso,  &  misura,' 
Venice,  1553,  8°,  246  ff. 

A  beautifully  printed  little  manual  of  the  fundamental  operations  of 
arithmetic.  The  '  Abbreviatvre  delle  monete,  Pefi,  &  mifure,  chi  fi  ufano 
in  Vinegia'  (f.  2)  is  helpful  to  students  of  the  history  of  sixteenth- 
century  arithmetic  and  to  all  who  are  interested  in  the  metrology  of  the 
countries  with  which  Venice  traded. 


258 


RARA  ARITHMETICA 


Fig,  123.   Title  page  of  manzoni 


MICHAEL  STIFEL.    Ed.  pr.  1553.    Konigsberg  i.  Pr.,  1553. 

See  p.  223. 
Title.    See  Fig.  124. 

Colophon.   'Gedruckt  zu  K6nigs-//berg  in  Preufse  durch  Alex- 
andrum // Behm   von  Luthomifl/ Voll //endet   am   dritten  tag 


PRINTl^)    1KK)KS  259 

dcfs  1  Icrbft-//m<^n^ts/ Als  niann  /.alt   nach //dcr  gcburt  vnfcrs 
lichen //hcrrn  Jcfu  Chrifti.//'  1554.'     (1'.  505.) 


§0  m^mgrptrglttprefilTttt 

ftj(tvniy*Uul<m  tin  j<jtr . 

Fig.  124.   Title  pac:k  ok  stikkl's  rudolff 

Description.  4°,  14.7  X  19.6  cm.,  the  text  being  9.9  X  16  cm. 
507  pp.  (4  blank,  12  unnumb.),  21-26  11.  Konigsberg  in  PreuCen, 
1553- 


26o  KARA   ARITHMETICA 

Editions.  Konigsberg  in  Preufien,  1553,  4°  (here  described); 
1 57 1.     See  also  p.  226. 

Although  the  book  is  nominally  an  algebra  (the  word  '  cofs  '  coming 
from  the  Italian  cosa, '  thing,'  meaning  the  unknown  quantity),  the  first 
part  of  the  book  is  devoted  entirely  to  arithmetic.  This  part  is  intended 
as  an  introduction  to  the  algebra  that  comes  later,  and  gives  the  funda- 
mental operations  as  treated  by  Rudolff,  with  Stifel's  commentary.  It 
is  entirely  theoretical,  and  it  naturally  leads  into  the  theory  of  irrational 
numbers,  which  constitutes  the  first  part  of  the  algebra. 

Other  works  0/1553.  Albert,  p.  180,  1534  ;  Boethius,  p.  27,  1488  ; 
Gemma,  p.  204,  1540;  Gyraldus,  p.  254,  1552  ;  Mariani,  p.  181,  1535  ; 
Morsianus,  p.  182,  1536;  Ringhieri,  p.  253, 155  i ;  Rudolff,  p.  152,  1526; 
Bernardu  Wojewddki,  <■  Algorithm,  to  jest  nauka  liczby,  po  polsku  na 
linijoch  uczyniony,'  Cracow. 

CLAUDE  DE  BOISSIERE.     Ed.  pr.  1554.  Paris,  1554. 

Claudius  Buxerius.  Born  in  the  diocese  of  Grenoble,  probably  c.  1500. 
He  also  wrote  on  poetry,  music,  and  astronomy.  For  his  Rythmomacliia  see 
p.  271. 

Title.    See  Fig.  125. 

Colophon.  '  Acheue  d'Imprimer  le  xiij. // iour  d'Octobre,// 
1554.'     (F.  74,  I.) 

Description.  4°,  10  X  14.8  cm.,  the  text  being  7.2  x  12.9  cm. 
2  ff.  unnumb.  +  73  numb.  =  75  ff.,  32  11.     Paris,  1554. 

Editiotis.  Paris,  1554,  8°(here  described);  ib.,  1563,  8°(p.  262). 

This  is  a  theoretical  work  in  two  '  livres,'  probably  written  for 
students  in  the  University  of  Paris.  It  is  the  second  printed  work  which 
I  have  noticed  that  carries  the  system  of  numeration  as  high  as  thousands 
of  quintillions,  '  Mille  de  Quintillions,'  although  Chuquet,  in  his  manu- 
script of  1484,  carries  it  to  '  nonyllions.'  De  la  Roche  (p.  128)  followed 
Chuquet  in  this  as  in  other  respects.  Boissiere's  plan  starts  with  the 
names  of  millions,  bimillions  (for  million  millions),  trimillions,  etc.  He 
then  says  that  to  avoid  ambiguity,  as  of  bimillions  for  two  millions,  these 
names  are  abridged  to  billion,  trillion,  etc.  He  moreover  numerates  in 
periods  of  six  figures  each,  as  in  England  at  present,  and  not  in  periods 
of  three  figures  as  is  now  the  custom  in  France  and  America. 

He  first  treats  of  the  classes  of  number  according  to  the  old  plan, 
distinguishing  between  digits  (1-9),  articles  (multiples  of  10),  and  com- 
posites (articles  +  digits),  of  which  names  we  still  use  the  digit.  Instead 
of  extending  the  fundamental  operations  to  six,  or  even  nine,  as  his 


PRINTED    HOOKS  261 

contemporaries  so  often  did,  he  limits  them  to  four,  as  we  do.  His  work 
in  addition  and  multiplication  is  substantially  like  ours,  but  his  '  sub- 
straction  '  is  as  follows  (1563  edition)  : 

Refte  939973901     ^• 

Debte  on  fomme  1840006503     A. 

Paye  on  a  Subftraire  900032602     ]'>. 

L'ART 

D'ARYTHME- 

TldVE     CONTENANT 

TOVTE  DIMENTION,    TRES- 
SINGVLIER  ET  COMMODE, 

tant  pour  Tare  militaire  que 
autrcs  calculations. 


Auec  priuilege 
du  Roy. 

Im^r'mia  ParU,par^nnet  Briere,a  I'cnfeignefiinH 

iehajiian ,  rue  da  Voriis . 

15    5   4- 

Fig.  125.   Title  page  of  hoissiere 


262  RARA   ARITHMETICA 

In  division  he  uses  the  galley  method  only.  Boissiere  is  one  of  the  first 
writers,  I  believe  the  first  in  France,  to  invert  the  divisor  in  the  division 
of  fractions,  as  Stifel  had  done  a  few  years  before  in  Germany.  Book  I 
closes  with  a  somewhat  extended  treatment  of  the  rule  of  three. 

The  second  book  relates  largely  to  mensuration,  figurate  numbers, 
roots,  and  the  mediaeval  proportions. 

Some  of  the  rules  are  in  verse.  Of  military  matters,  referred  to  in 
the  title  page,  there  is  little  mention. 

CLAUDE  DE  BOISSIERE.     Ed.  pr.  1554.  Paris,  1563. 

See  p.  260. 

Title.  This  is  practically  identical  with  that  of  the  1554  edi- 
tion, except  for  the  following  :  '  Reueu  &  augmente  par  Lucas 
Trembley  Parifien,//profeffeur  des  Mathematiques.// A  Paris,// 
Pour  Guillaume  Cauellat,  a  I'enfeigne  de  la  Poulle//graffe,  deuant 
le  college  de  Cambray.//  1563.'    (F.  i,  r.) 

Description.  8°,  10.2  X  i6.8cm.,thetext  being  6.8  X  13  cm.  3ff. 
blank  +  i  unnumb.  +  71  numb.  =  75  ff.,  25-30  11.     Paris,  1563. 

See  p.  260.  Two  other  books  are  bound  with  this  :  '  La  Declara// 
tion  et  Vsage  de  //  L'inltrument  nomme  Canomettre,//  Par  G.  des 
Bordes,  Gentilho-//  me  bordelois,  profeffeur  ez  //  Mathematiques.//  .  .  . 
1570.'  and  '  Vsage  //  Dv  //  Compas  //  de  //  Proportion.//  Par  D.  Hen- 
rion,  Mathem.//  .  .  .  M.  DC.  XVIIL' 

JOACHIM   CAMERARIUS.     Ed.  pr.  1554.    Deventer,  1667. 

Born  at  Bamberg,  April  12,  1500;  died  at  Leipzig,  April  17,  1574.  The 
office  of  chamberlain  (Kammermeister)  to  the  Prince-Bishop  of  Bamberg 
being  hereditary  in  the  family  of  Liebhard,  he  took  the  Latin  name  of 
Camerariu.s.  He  was  a  distinguished  classicist,  a  friend  of  Melanchthon, 
and  a  professor  at  Tubingen  and  Leipzig.  Of  the  various  commentaries  on 
Nicomachus,  his  was  the  most  important  of  the  Renaissance. 

Title.  '  Explicatio  //  loachimi  Camerarii  //  Papebergenfis//  in 
dvos  libros//Nicomachi  Geraseni//Pythagorei //Deductionis// 
Ad  Scientiam  Numerorum.//  Et  Notse  //  Samuelis  Tennulii  //  in 
//Arithmeticam  // Jamblichi  Chalcidensis.//Daventrise.//Typis 
Wilhelmi  Wier,  CID  I3CLXVII.'     (P.  i.) 

Description.  4°,  14.7  X  19.8  cm.,  the  text  being  10.3  X  14 
cm.     2  pp.  unnumb.  +  239  numb.  =  241  pp.,  34  11.     Bound  with 


PRINTED    BOOKS  263 

the  commentary  of  lamblichus  on  Nicomachus  (jx  188,  1538). 
Deventer,  1667. 

Editions.  The  first  edition  was  Augsburg,  15  54,  8°.  There 
was  also  an  edition  pubUshed  in  1569. 

Camerarius  also  wrote  a  work  entitled  '  De  logislica,'  published 
at  Augsburg  in  i  554  ;  ib..  i  557,  8° ;  Leipzig  (?),  1569,  8°.  See 
also  p.  257,  1552. 

A  commentary  on  the  theoretical  work  of  Nicomachus  (p.  i86). 

Other  works  0/1554.  Albert,  p.  180,  1534;  Bseda,  p.  131,  1521; 
Buteo,  p.  292,  1559  ;  Euclid,  p.  238,  1545  ;  Finaeus,  p.  160,  1530-32  ; 
Fischer  (Piscator)  p.  247,  1549;  Glareanus,  p.  192,  1539;  Huswirt, 
p.  74,  1501  ;  Nicomachus,  p.  186,  1538 ;  Peletier,  p.  245,  1549  ;  Psellus, 
p.  168,  1532  ;  Riese,  p.  139,  1522  ;  Stigelius,  p.  249,  1549  ;  Tagliente, 
p.  114,  1515  ;  Barth.  Barchi,  'Tariffe  della  valuta  di  tutte  le  monete,' 
Mantua,  4°  (hardly  an  arithmetic)  ;  Claudio  Bertholio,  '  De  numerandi 
ratione  aphorismi,'  Paris,  8°  ;  Ian  Gentil  (Vander  Schuere,  in  his  1634 
edition,  f.  201  — see  p.  424  —  refers  to  his  arithmetic  as  published  at 
Paris,  1554);  Caspar  Hiitzler,  '  Eyn  behende  und  Kunstrik  Rekens- 
boeck  op.  der  Linien  und  Tziferen,'  I.iibeck,  8°. 

PETRUS  RAMUS.     Ed.  pr.  1555.  Paris,  1555. 

Pierre  de  la  Ramee.  Bom  at  Cuth,  Vermandois,  15 15;  killed  at  Paris 
the  night  of  August  24-25,  1572,  in  the  Massacre  of  St.  Bartholomew.  lie 
lectured  on  philosophy  at  Paris,  and  for  a  short  time  at  Heidelberg.  Me 
wrote  on  arithmetic,  geometry,  optics,  and  mathematics  in  general. 

Title.    See  Fig.  126. 

Colophon.  '  P.  Rami  Eloqventias  et  Phi-//losophi:E  professoris 
//  Regij  Arithmeticae,//  Finis.'    (P.  1 1 1 .) 

Description.  4°,  13.8  X  19.  i  cm.,  the  text  being  9. 5  x  17.2  cm. 
128  pp.  (2  blank,  16  unnumb.),  26-27  11.     Paris,  1555. 

Editions.  Paris,  1555,  4°  (here  described);  ib.,i557,8°;  Basel, 
1567;  Paris,  1584.  See  also  Gleitsman,  1600  (p.  427),  and  the 
'  Libri  Duo'  of  Ramus,  1569  (p.  330). 

This  arithmetic  was  popular  in  the  Latin  schools  for  half  a  century. 
It  is  theoretical,  consisting  largely  of  definitions,  extracts  from  the  Greek 
writers,  a  little  work  on  the  fundamental  operations,  and  the  medioeval 
theory  of  ratios.  Ramus  had  not  the  faculty  of  putting  together  a  text- 
book that  should  be  a  rival  to  that  of  (rcmma  Frisius. 


264  RARA   ARITHMETICA 


P.  Rami,  cloqucntiae 

ET    PHILOSOPHIC^    PROFES- 

SORIS     REGII,ARITH- 

meticce  libri  trcs. 


Carolum  Lothar'mgum  Cardinakm, 


P  A  R  T  S 1 1  S, 

Apud  Andream  'W^echelum,  Tub  Pcgafb,in 
vicoBellouacO:,      ^nno  Salutif, 

M  5  5- 

Cum  priuilegio  Regis. 

Fig.  126.    Title  page  of  ramus 


PRINTED    IJOOKS  265 

JACOBUS  MICYLLUS.     Ed.  pr.  1555.  Basel,  1555. 

MoLTZER.   Born  at  Strasburg,  April  6,  1503;  died  at  Heidelberg  (?),  January 
28,  1558.     He  was  a  well-known  classical  scholar. 

Title.    See  Fig.  127. 

Arichmctjc^ 

LOGISTICAE      HBRI 

duo ,  cx  di'ucrfis  cius  artis  fcri^ 

ptoribus  coIIe<Sij  &  excmplis 

plurimis,  ijfdein'c^  utilifs^ 

nupcrilluftrati: 


PER     lACOBVMMU 


Cum  gratia  K priuilf gio  Imperlali 
ad  decenniiim. 


BASILEAE,    PER   lOAM* 

urn  O^ormm. 

Fig.  127.    Title  page  of  iMicyllus 

Colophon.  '  Basileae,  ex  ofificina//IoannisOporini,  Anno  Salutis 
humanae//M.  D.  LV.  Menfe  Martio.'    (P.  319.) 

Description.  8°,  9.3  X  15.2  cm.,  the  text  being  7.8  x  1 1.3  cm. 
320  pp.  (3  blank,  23  unnumb.),  25-27  11.     Basel,  1555. 


266  RARA   ARITHMETICA 

Editions.  There  was  no  other  edition.  Treutlein's  statement 
{AbhandlitJigcn,  I,  15)  that  the  book  was  pubhshed  in  Heidelberg 
is  incorrect.  It  probably  came  from  the  fact  that  the  '  Epistola 
Nvncvpatoria'  is  dated  '  Heidelbergae,  10  Calend.  Nouemb.  Anno 
Domini  1553.' 

The  book  was  written  for  the  classical  schools.  Although  Micyllus 
gives  the  fundamental  operations  in  a  practical  manner,  the  latter  part 
of  his  book  is  theoretical,  presenting  some  of  the  ancient  arithmetic  in 
the  Greek  language.  The  work  includes  an  unusually  complete  treat- 
ment of  sexagesimals,  '  De  partibvs  Aftronomicis,  &  earum  fupputa- 
tione '  (p.  201),  and  a  chapter  on  the  computus  (see  p.  7),  '  De 
Temporvm  svppvtatione,  qva  Ecclefiaftici  utuntur.'  The  latter  shows 
that  the  name  '  Computus  '  had  become  unpopular,  for  it  begins  with  the 
following  statement :  '  Est  &  temporum  fupputatio  quaedam,  quem  Com- 
putum  Ecclefiafticum,  barbaro  nomine,  uocant.'  (P.  269.)  The  follow- 
ing is  an  example  of  his  applied  problems  :  '  Scribit  Plinius,  Alexandriam 
a  Rhodo  diftare  583000  pafsuum.  Cupio  autem  fcire,  quot  ftadia  ijdem 
pafsus  efhciant.  Diuido  igitur  583000  per  125.' 

MICHAEL  NEANDER.     Ed.  pr.  1555.  Basel,  1555. 

Born  in  the  Joachimsthal,  April  3,  1529;  died  at  Jena,  October  23,  1581. 
He  was  professor  of  mathematics,  Greek,  and  medicine,  in  the  university  of 
Jena.     He  wrote  on  physics  and  cosmography. 

Title.    See  Fig.  128. 

Colophon.  '  Basileae,  ex  Officina  //  loannis  Oporini,  Anno  // 
Salutis  humanse  M.  D.  LV.  Men-//fe  lanuario.'     (P.  119.) 

Description.  4°,  13.9  X  19.5  cm.,  the  text  being  9.1  x  15.7  cm. 
104  pp.  numb.  +  15  unnumb.  +  i  blank  =  120  pp.,  25-29  11. 
Basel,  1555. 

Editions.  There  was  no  other  edition.  The  '  Epistola  Nvncv- 
patoria '  is  dated  '  Die  Pafchalis,  Anno  LI  1 1 1,'  but  the  book  was 
printed,  as  the  colophon  shows,  in  1555. 

This  is  a  historical  treatise  on  Greek,  Egyptian,  Roman,  Arabic, 
and  mediaeval  European  weights  and  measures,  and  is  particularly  inter- 
esting because  of  the  symbolism  which  it  contains.  The  origin  of  our 
present  apothecary's  symbols  of  measure  is  seen  in  the  ancient  Roman 
abbreviations.  The  text  is  mostly  in  Latin,  but  eight  pages  of  '  VaX-qvov 
TTf.pL  Mfrpojv  Koi  Sra^/Awv'  are  in  Greek. 


SYNOTIE 

MENSVRARVM    ET    PON- 

DERVM,         POND  E  RATI  ONI  sO 

que  mcnfurabilium  fecundum  Romanos, 

Athenienfes ,  yuu^xcyjQ ,  msu  iTmoiiT^cvs ,  ex 

prseilantiTsimis  authoribus  huius  generis 

cotrafl:a,opera  MiCHAELis   NEAN^ 

DRi  exVallcIoacfiimica, 

Anno  M  D  Liiiu 


ACCESSERVNT    ETIAM,    C^VAE    APVD 

QAlenum  hd^lemis  cxtabm  deponderumcrmenfuriirum  rdtione  U€e 

hmcnter  ieprauAtdymnc  Grace  cr  Latinc  multo  cotY(^ior4f 

€iufdm  MICHAELIS    NEAN« 

Item  rcrum  dC  ucrborum  in  his  omnibus 
niemorabilium  Index. 


LEVITICI     XIX. 


EASILEAE,      PER      1  O  A  .^^ 

nem  Opormum, 

Fic.  128.    Title  page  of  neander 


268 


RARA   ARITHMETICA 


ANTOINE  CATHALAN(?).     Ed.  pr.  1555. 


Paris,  1556. 


A   French  arithmetician.    The  author's  name  does  not   appear  in   this 
edition.    A  work  with  the  same  title  is  assigned  to  Cathalan,  Lyons,  1555. 

Title.    See  Fig.  129. 


LARITHMETIQVE 

ET    MANIERE     DE    >\PPRENDR.E 

a  Chijfur  ^  compter  f>ar  la  plume  ^  pat 

les^efiii  en  nomhre  ent'ter  ^  yam  • 

pu,facile  A  apprettdre,^  tref- 

utileatoutes  rrens, 

Denouueau  reueuc 
Sc  corrigee. 

xy^laquellefont  adioitflees  phtfetm  quefiions 
^.  exemples  pour  fair  e  lafaenceplui 
facile^^  plus  k^me  a  (om- 
prendre. 


A     PARIS. 

Par  lehan  RucIIejJemournnt  en  la  Rue 

fain  (ft  Iacqucs,a  Tenfcigne  dc 

Ja  queue  de  Regard. 

Fig.  129.    Title  page  of  the   1556  cathalan 


Description.      Small  8°,  7.3  X  11.4  cm.,  the  text  being  5.3  X 
9.5  cm.     79  ff.  numb.  +  i  unnumb.  =  80  ff.,  27  11.    Paris,  1556. 


PRINTED   BOOKS  269 

Editions.  Lyons,  1555,  16°;  Paris,  1556,  sm.  8°  (here 
described). 

This  is  a  very  good  httlc  primer  of  algorism  for  the  lime.  Concern- 
ing this  work  the  author  says  :  '  Lequel  art  trouua  premierement  vn 
philofophe  d'Arabie,  nomme  Algus.  Dont  cefte  fcience  prent  fon  nom 
d'Algorilme  '  (f.  3).  In  multiphcation  the  gelosia  arrangement  is  given 
as  a  third  method,  under  the  name  '  Mvltiphcation  i)er  quarreaux.'  In 
division  only  the  galley  form  appears.  The  '  premier  liure  '  covers  the 
four  fundamental  operations  with  integers,  and  a  brief  treatment  of  pro- 
gressions. The  '  fecond  liure'  relates  to  the  use  of  counters  ('gectz'), 
which  the  author  esteems  so  highly  as  to  say  :  '  ICt  note  que  cefte  efpece 
de  addition  eft  plus  vtile  &  facile  aux  gectz  que  aux  chifres  ; '  '  il 
eft  facile  par  les  gectz,  &  difficile  par  les  chifres.'  The  last  part  of  the 
book  is  devoted  to  such  standard  problems  as  the  testament,  the  pipes 
filling  the  cistern,  and  the  broken  eggs,  and  to  applications  like  ex- 
change and  pasturage. 

LODOICO  BAEZA.     Ed.  pr.  1555.  Paris,  1555. 

A  Spanish  scholar  of  the  sixteenth  century. 

Title.    See  Fig.  130. 

Colophon.  '  Excudebat  Benedictus  Preuotius,  via // Fremen- 
tella,  ad  infigne  ftellas  aureas.'    (F.  68,  r.) 

Description.  8°,  10.3  X  15.6  cm.,  the  te.xt  being  7.8  X  13  cm. 
2  ff.  unnumb.  ^66  numb.  =-(i'$>  ff.,  26-27  ^l-     P^"s,  1555- 

Editions.  This  rare  first  edition  of  Baeza's  arithmetic  is  sel- 
dom found  in  libraries.  The  second  edition  (1556),  8°,  is,  how- 
ever, the  same  impression  with  the  title  page  changed. 

The  book  is  entirely  theoretical,  making  much  of  the  classification 
of  numbers  and  of  the  ancient  ratio  systems.  It  is  in  Latin  with  nu- 
merous Greek  quotations. 

Other  works  of  1555.  Euclid,  p.  237,  1545  ;  Finaeus,  p.  163, 
1530-32  ;  Gemma,  p.  200,  1540  ;  Glareanus,  p.  192,  1539  ;  Herman 
Giilfferich,  p.  257,  1552;  Mariani,  p.  181,  1535  ;  Ringhieri,  p.  253, 
155 1  ;  Valturius,  p.  10,  1472  ;  Yciar,  p.  249,  1549  ;  Jacob  Cuno,  'Arith- 
metica,'  Wittenberg,  8° ;  Taddeo  Duni,  '  Liber  de  arithmetica,'  Basel, 
4°  ;  Melchior  Goldammer,  'Arithmetica  Pratica,'  Wittenberg,  8°  ;  Jero- 
nimo  de  Valencia,  'Arte  de  computo,'  in  Santaella's  'Vocabularium 
ecclesiasticum,'  a  separate  edition  appearing  at  Saragossa  in  1601.  (See 
also  p.  140,  1523.) 


270 


RARA   ARITHMETICA 


N  V  M  E  R  A  N- 

DI     DOCTRINA 

PR  AE  CLARA        METHODO 

expojita,  in  qua  hreuiter  continentur,  ^ 
exponmtur  aperte  ea,quxex  ynmrfi  ^~ 
rithmetka  Junt  ad  yfim  potiora, 

^uthore  Lodoico  Bae:^, 


L  V  T  E  T  I   AF, 

^pudGulielmum  Cauellat  yftibphigui  Galli 
nnjcx  aduerfo  coilegij  Cameracenfis, 
J     5    5    5. 

Fig.  130.  Title  page  of  baeza 


PRINTED    BOOKS  271 

CLAUDE  DE  BOISSIERE.     Ed.  1)1.1556.  Paris,  1556. 

See  p.  260. 

Title.    Sec  Pi<i.  131. 

Dcscriptioit.  8°,  10.4  X  16.6  cm.,  the  text  bein<;  6.7  x  13.1 
cm.  1  f.  unnumb.  +  52  numb.  +  i  blank  =  54  ff.,  26-27  11. 
Paris,    1556. 

Editions.  Paris,  1556,  8°  (here  described);  French  edition, 
ib.,  1556,  8°. 

Of  the  three  standard  treatises  on  the  ancient  number  game  of 
Rv  thmomachia  mentioned  in  this  list,  the  others  being  the  one  of  1496 
of  uncertain  authorship  and  Barozzi's  work  of  1572,  this  is  the  clearest. 
It  describes  very  carefully  the  checkerboard  on  which  the  game  is 
played,  the  nature  of  the  calculi  used,  and  the  general  mode  of  procedure. 
Moreover,  it  is  profusely  illustrated  (Fig.  132),  which  adds  much  to  the 
value  of  the  book.  The  game  was  connected  with  the  mediaeval  number 
classifications  and  ratios,  and  could  never  have  been  understood  by  any 
save  those  who  were  well  educated  in  the  ancient  theoretical  arithmetic. 

GALLUS  SPANLIN.     Ed.  pr.  I556(.?).  Nurnberg,  1566. 

An  Ulm  Rechenmeister,  as  he  describes  himself  on  the  title  page. 

Title.    See  Fig.  133. 

Colophon.  'Gedruckt  zu  Nurmberg//durch  Chriftoff  //  Heufz- 
ler.'    (P.  382.) 

Description.  8°,  9.2  x  14.7  cm.,  the  text  being  7.1  X  1 1.5  cm. 
15  pp.  unnumb.  +  4  blank  +  365  numb.  =  384  pp.,  21-25  11- 
Niirnberg,    1566. 

Editions.  The  dedication  to  '  Den  Edlen  Ehrnveften/  Fiirfich- 
tigen/ Erfamen  vnd  Weyfen  Herren/Eltern  Burgermeiftern  vnd 
Rath/des  heyligen  Reichs  Stadt  Ulm'  is  dated  '3  Julij/Anno 
Chrifti/  1556,'  so  that  this  is  possibly  the  date  of  the  first  edi- 
tion.   I  know  of  no  other  edition  except  this  of  1566. 

The  author  devotes  twenty-one  pages  to  '  Rechnung  auff  der  Linien,' 
this  counter  reckoning  being  still  the  popular  method  at  the  time  he 
wrote.  He  then  (p.  24)  takes  up  the  processes  with  Arabic  numerals, 
at  first  using  a  few  abstract  numbers,  but  soon,  as  was  the  custom, 
introducing  many  practical  applications.  On  the  whole  the  book  may 
be  said  to  be  a  rather  good  exponent  of  Riese's  school. 


272 


KARA   ARITHMETICA 


NOBILISSI- 

MVS    ET    ANTIQVISSt- 

mus  ludusPythagoreus  fqui  Rythmo 
machia  nominatur)  in  vrilitatem  &  re- 
laxationem  ftudioforum  comparatus 
ad  veram  Sc  facilem  proprietatem  & 
rationem  numerorum  aflequendam, 
nunc  radem  per  Claudium  Buxerium 
Uelphinatcm  illuftratus. 


L  V  T  E  T  I  AE, 

^pudGulielmum  Cauellat.fHhfmgm  Gal^ 
limiiCX  adaerfo  coUegij  Camentcenfis. 

^hdCiM  O*  calculi  yxneum  in  PtiUtio, 
aptidloanncm  Genu  I . 

M  5  ^ 

CVM     PRl  V  ILEGIO     REGIS. 

Fig.  131.   Title  page  of  the   1556   Boissiere 


PRINTED    BOOKS 


273 


esfiii 


13^ 


BAAAABS 


Fig.  132.    From  boissiere's  Rythmomachia 


274 


RARA  ARITHMETICA 


vn^^f^ctnivozm(An 
nicgcfebcrtt 

mcyflci!  inn  X)Im  /  5ttb:ii^ 
€yrftc&flmjo€«pft( 

Fig.  133.   Title  page  of  spanlin 


■;^?«^1 


PRINTED    BOOKS  275 

DOMENICO  DELFINO.     Ed.  i)r.  1556.  Venice,  1565. 

A  sixteenth-century  Venetian.  He  was  born  of  a  noble  family,  which  had 
produced  a  doge,  a  general  of  the  Camaldoli,  and  other  men  of  prominence, 
and  which  was  later  to  produce  a  cardinal  and  a  captain-general  of  the 
Venetian  fleet. 

Title.    See  Fig.  134. 

Description.  8°,  9.8  X  14.4  cm.,  the  text  being  8.5  x  12.7  cm. 
416  pp.  (56  unnumb.),  26  11.    Venice,  1565. 

Editions.  Venice,  1556,  4°;  ib.,  1565,  8°  (here  described); 
ib.,  1568,  8°;  ib.,  1584,  8°,  and  later.  The  title  page  bears  the 
date  1565  as  stated,  and  the  dedicatory  letter  '  Al  Mag.^o  et 
Ecc.™°  Sig.  Nicolo  Crasso,'  signed  by  Ludouico  Dolce,  is  dated 
*  In  Venegia  a  VII.  di  Febraio.  M  D  LXIIII,'  1565  new  style. 
The  author's  preface,  '  Intentione  dell'  avtore,'  which  follows, 
bears  no  date,  but  succeeding  this  there  is  a  letter  '  All'Illvstris- 
simo  et  Reverendissimo  Signer,  II  S.  Cristoforo  Madrvccio, 
Cardinal  di  Trento,  Governator  Meritissimo  di  Milano,'  signed 
by  F.  Nicolo  Croce,  and  dated  *  Di  Vinetia  il  di  primo  di  Settem- 
bre.  M  D  L  VI.'  There  is  no  colophon  to  fix  the  date  more 
accurately,  unless  it  appeared  on  some  page  after  360,  with  which 
this  copy  terminates. 

The  book,  a  '  summary  of  all  the  sciences,'  devotes  a  brief  chapter 
to  the  Boethian  arithmetic  (pp.  43-48)  :  '  Dell'  Aritmetica,  de'  svio  // 
inuentori,  utilita,  modo,  &  altri  fecre  //  ti  Cap.  III.'  The  author  is  highly 
lauded  in  Croce's  letter  as  of  noble  family,  '  Illuftre  per  lo  splendor 
del  fangue,  non  dimeno  uia  piu  per  I'ornamento  e  per  la  gloria  della 
dottrina.'  The  Delfino  family  was  at  that  period  very  prominent  in 
Venice. 

NICOLO  TARTAGLIA.     Ed.  pr.  1556.  Venice,  1556. 

Tartalea.  Bom  at  Brescia,  c.  1506;  died  at  Venice  in  1559.  He  was 
one  of  the  best  mathematicians  of  his  time,  and  was  the  first  to  give  a  gen- 
eral solution  of  the  cubic  equation. 

Title.    See  Fig.  135. 

Description.  Fol.,  21.1  X  29.  i  cm.,  the  text  being  13.8  X  23.6 
cm.    285  ff.  (3  blank,  6  unnumb.),  52-56  11.    Venice,  1556. 

Editions.  Tartaglia's  works  include  the  '  Nova  Scientia,' 
chiefly  algebraic,  editions  of  Euclid,  Archimedes,  and  Jordanus, 


276 


KARA   ARITHMETICA 


S    C    I    E    N    Z    E, 


D_EL    M^G-N  IFI  CO    M  E  S  <:  EJ^ 


Domenico  Deljino  y  nobile  Vmitiano , 


D AL  OVALE   SI  POSSONO   IMPARA^ 


moIte~cole  appartenenti  al  uiuere  humaiio^ 
&  alia  cogmnon  di  D  1  o . 


Con  U  lattola,  gr  le  FojhUe  delle  cofe  pm  notahdi 


G  1  O  L  I  TO    D  E'    FERRAR  I. 
M       n       T      T     V 

Fig.  134.   Title  page  of  delfino 


H/f 


LA    PRIMA    PARTE     DEL 

GENERAL     TRATTATO     DI      NV* 

MERI,  ET  MISVRE  DI  NICOLO   TARTAGLIA, 

NELLAaVALE     IN     DIECISETTE 

UBRI  SI  DICHIARA  TVTTI  GLI  ATTI  OPEU;ATIV/, 

PRATICHE,   ET   REGOLE   NECES5ARIE  NON  SOLA' 

incntc  in  tutra  I'arte  ntgoiijria,&  mcrcantile.ma  anclior  m  ognialtra 

art«,fckntia,oucr  difciplina,  douc  intcrucnglii  il  calculo. 


CON   LI  SVOI    PRIVILEGir. 

In  T^inegia  per  Ciirtio  Troiam  de  i  Mauo, 

M    D    LJy^I. 

Y\v,.  135.   Title  page  ok  taktaglia 


278  RARA   ARITHMETICA 

the  '  Quesiti  et  inventioni '  and  '  Regola  generale,'  chiefly  phys- 
ical, and  the  '  General  Trattato  '  containing  the  arithmetic.  Of 
the  arithmetic  there  were  editions  in  Venice,  1556  and  1560, 
4°  (?) ;  ib.,  1592-93,  4°  (p.  279)  ;  Paris,  1578,  8°  (below) ;  Ant- 
werp, 1578,  8°.  There  was  also  a  '  Scelta  di  Abbaco  ridotto  dal 
famosissimo  Nicolo  Tartaglia,'  published  at  Venice  in  1596. 

The  entire  work  consists  of  six  volumes,  bound  usually  in  three,  and 
the  publication  extended  over  a  period  of  five  years,  from  1556  to  1560 
inclusive.  This  volume  contains  the  arithmetic,  the  most  scholarly  con- 
tribution to  the  subject  that  appeared  in  the  sixteenth  century.  It  is 
more  elaborate  than  the  treadse  of  Paciuolo  (p.  54,  1494),  and  hke 
that  work  it  enters  into  the  various  minute  details  of  the  operations 
and  commercial  rules  of  the  Italian  arithmeticians.  For  example,  Tar- 
taglia gives  seven  methods  for  the  multiplication  of  integers,  and  four 
for  division  ;  he  enters  very  fully  into  the  discussion  of  denominate 
numbers  and  exchange  ;  and  his  treatment  of  such  rules  as  '  Regula  de 
tri '  is  unusually  elaborate.  Indeed,  there  is  no  other  treatise  that  gives 
as  much  information  concerning  the  arithmetic  of  the  sixteenth  cen- 
tury, either  as  to  theory  or  application.  The  life  of  the  people,  the 
customs  of  the  merchants,  the  struggles  to  improve  arithmetic,  are  all 
set  forth  by  Tartaglia  in  an  extended  but  interesting  fashion. 

NICOLO  TARTAGLIA.     Ed.  pr.  1556.  Paris,  1578. 

See  p.  275. 

Title.  '  L'Arithmetiqve  //  de  Nicolas  //  Tartaglia  Brescian,// 
grand  mathematicien,//et  prince  des  practiciens.// Diuifee  en 
deux  parties.//  La  declaration  fe  verra  en  la  page  fuyuante.// 
Recueillie,  &  traduite  d'ltalien  en  Francois,  par // Gvillavme 
Gosselin  de  C8en.//Auec  toutes  les  demonftrations  Mathema- 
tiques  :  &  plufieurs  in-//uentions  dudit  Gosselin,  efparfes  chacune 
//en  fon  lieu.// A  tres-Illustre  &  Vertueufe  Princeffe  Mar-// 
gverite  de  France,  Royne  de  Nauarre.// Premiere  Partie.//A 
Paris,// Chez  Gilles  Beys,  rue  S.  lacques,  au  Lis  blanc.//  1578. 
//Avec  Privilege  dv  Roy.'     (F.  i,  r.) 

Description.  8°,  10.6  X  16.6  cm.,  the  text  being  7.5  x  13.4 
cm.  286  ff.  (152  in  part  I,  134  in  part  II,  28  being  unnumb.), 
27-32  11.     Paris,  1578. 


PRINTED    BOOKS  279 

See  p.  278.  The  '  Seconde  Partie  '  follows  f.  numb.  136  of  the  first 
part,  and  is  of  the  same  date.  The  Privilege  follows  f.  numb.  122  of 
the  second  part,  and  is  dated  '  Paris  le  17.  Septembre.  1577.' 

This  is  merely  a  French  translation  of  the  work  already  described 
with  notes  by  Gosselin.  These  notes  are,  rather  naively,  printed  in 
more  prominent  type  than  the  original  text,  but  are  generally  of  little 
value.  They  cover  such  points  as  multiplying  by  or  dividing  numbers 
ending  in  zero,  they  amplify  certain  demonstrations  (as  in  the  division 
of  fractions),  and  they  adapt  the  commercial  chapters  to  French  usage. 

NICOLO  TARTAGLIA.     Ed.  pr.  1556.  Venice,  1592. 

See  p.  275. 

Title.  '  Tvtte  I'opere  //  d'arithmetica  //  del  famosissimo  // 
NicoloTartaglia.// Nolle  qvale  in  XVII.  libri  con  varie  prove,// 
&  ragioni,  moftrafi  ogni  prattica  naturale,  &  artificiale;  i  modo,  & 
le  regole  da//gli  Antichi,  &  Moderni  vfate  nell'  arte  mercantile; 
&  cue  interuiene  calcolo,//peri,  denari,  tariffe,  calmeri,  baratti, 
cambi  di  banchieri,  e  di  fiere,  faldi,  fconti,//giuochi,  traffico  di 
compagnie,  compre,  vendite,  portar  mercantie  da  un  paefe  // 
all'altro,  conuertir  monete,  congiungimento  di  metalli,  &  opere 
de'zecchieri.//Sopra  le  qual  cofe  tutte,  formanfi  bclliffimi  que- 
fiti,  &  fi  fciolgono  le  dififi-//colta,  con  vgual  chiarezza,  &  dili- 
genza,  per  vtile  rileuato  de  i  mercanti,  &  te-//forieri,  a  Capitani, 
e  Matematici,  &  Aftrologhi,  &c.  //  Parte  Prima.//  Con  Privilegio. 
// In  Venetia,// Air Infegna  del  Leone.    M.  D.  XCIL'    (F.  i,  r.) 

Description.  4°,  13.6  X  18.8  cm.,  the  text  being  1 1.5  X  18  cm. 
479  ff.  (204  in  part  I,  275  in  part  II,  8  unnumb.),  40  11.  Venice, 
1592.    Riccardi  says  that  it  was  not  completed  imtil  1593. 

Editions.    See  p.  278. 

This  is  substantially  the  same  as  the  first  volume  of  the  '  General 
Trattato  '  of  1556  (p.  275),  except  that  it  is  in  (juarto  instead  of  folio. 

ORONTIUS  FINAEUS.     Ed.  pr.  1556.  Paris,  1556. 

See  p.  160. 

Title.    See  Fig.  136. 

Description.  Fob,  19.6  X  28.5  cm.,  the  text  being  13  X  21.4 
cm.    6ff.  unnumb.  +  136  numb.  =  142  ff.,  15-34  11.    Paris,  1556. 


ORONTIIFINAEI, 

DELPHINATIS.RE- 

GII    MATHEMATI- 
CARVM    PRO- 

t  ES  so  RIS, 

De  rebus  mathemacicis, 
hadenus  defidcracis, 

Libri  nil. 


C  Quibus  inter  caetcra,  Circuli  quadratura  Centum 

inocIis,&fupra,pcrcundem  Orontium 

rcccntercxcogitatis,dcmonftratur. 


LVTETIAE    PARISIORVM, 
Anno  Chrifti  Seruatoris, 
M.   D.   LVI. 


ExofficinaMichaelis  VafcoJfaniiUia  lacobara 
adinfigneFontis. 

Cum  Pnuilcgio  Regis. 

Fig.  136.  Title  page  of  the  1556  finaeus 


PRINTKI)    HOOKS  281 

Editions.  There  was  no  other  etlilion.  For  liis  arithmetic, 
see  J).  160. 

Akhough  chiefly  on  geometry,  this  work  is  included  in  the  Hst  of 
arithmetics  because  its  treatment  of  proportion  is  more  arithi-netical 
than  that  of  luichd  and  his  followers.  ( Kf .  25-29.)  There  was,  how- 
ever, nothing  original  in  the  work  of  Finaeus. 

VALENTIN   NAROD.     Ed.  pr.  1556.  Cologne,  1556. 

Naibod,  Naiboda.  Born  at  Cologne;  died  at  Padua,  March  3,  1593.  He 
was  for  a  time  professor  of  mathematics  at  Cologne.  lie  wrote  on  astronomy 
as  well  as  arithmetic. 

Title.    See  Fig.  137. 

Colophon.    '  Coloniae,  Typis  Iohannis//Bathenij.'    (F.  100,  r.) 
Description.    8°,  9.5  X  14.9  cm.,  the  text  being  6.3  X  1 1.9  cm. 
8  ff.  unnumb.  +  92  numb.  =  100  ff.,  28  11.     Cologne,  1556. 
Editions.    There  was  no  other  edition. 

This  is  a  I^tin  work,  with  occasional  Greek  passages,  and  was  written 
for  the  classical  schools  of  Germany.  Although  assuming  to  be  a  prac- 
tical treatise  on  calculating,  it  so  lacks  the  merit  of  brevity  as  to  be 
unfitted  for  use  as  a  textbook.  It  has  few  problems,  and  these  are  of 
no  commercial  value.  In  his  desire  to  exalt  the  classical  learning,  Nabod, 
like  other  Latin  writers  of  his  time,  assigns  the  Arab-Hindu  numerals  to 
the  Pythagoreans. 

VALENTIN   MENHER  de  Kempten. 

Ed.  pr.  1556.  Antwerp,  1556. 

See  p.  249. 

Title.    See  Fig.  138. 

Colophon.  '  Imprime  en  Anuers  par  Ian  Loe  //I'An  de  noftre 
Seigneur  1556 //le  20  iour  d'Auril.'     (F.  184,  v.) 

Description.  8°,  9.8  x  14.9  cm.,  the  te.xt  being  7.8  x  13.2  cm. 
184  ff.  unnumb.,  27-28  11.    Antwerp,  1556. 

Editions.    See  p.  250. 

This  is  usually  mentioned  as  the  second  of  Menher's  arithmetics.  It 
is,  however,  the  same  as  the  work  described  on  p.  249,  1550,  save  for  a 
few  minor  changes. 


282  KARA   ARITHMETICA 


VALENTI- 

NI    NABOD I    D  E 

CALCVLATORIA        NV- 

merorumque  naturaSe^io- 

nes  quatuor. 

CLMSJSSIMVM  ri<^M 

GASPARVM    DOVCIVM 

fLORENTINVM,  CAESAREAE 

Maieftatis  Confiliarium. 


COLONIAE     AGRIPPINAE, 

Apud  hseredes  Amoldi  Birck- 
mannl  1556. 

Fig.  137.    Title  page  of  nabob 


PRINT]^:i)    1U)()KS 


2«3 


ARITHMETI 

QJE   SECONDE   PAR 

M.  VALENTIN  MENNHER 

de  Kempten. 


Auec  grace  &  priuilege  de  I'Em- 
pereur  pour  quaere  ans. 

Fig.  138.    Title  pack  ok  menher 


284  RARA   ARITHMETICA 

PIERRE  FORCADEL.     Ed.  pr.  1556-57.       Paris,  1556-57. 

Born  at  Beziers  ;  died  at  Paris  in  1574.  He  was  (1560)  profe.ssor  of  mathe- 
matics in  the  College  Royale,  Paris.  He  also  wrote  on  astronomy,  and  trans- 
lated the  works  of  several  Greek  mathematicians. 

Title.  The  title  page  of  this  copy  of  Book  I  is  missing.  The 
title  page  of  Book  II  reads  as  follows  : 

'  Le  //  Second  Livre  de  // 1' Arithmetiqve  de  P.  For-//cadel  de 
Beziers.//  Av  qvel  seront  declarees  les// fractions  vulgaires,  auec 
leurs  demonftrations,  par  les  //  quantitez  continues,  &  premieres 
caufes  des  egaliffemens // de  I'Algebre.// Le  tout  nouuellement 
inuente  par  I'auteur.//  (Woodcut  with  motto  :  In  Pingvi  Gallina.) 
A  Paris,// Chez  Guillaume  Cauellat,  a  I'enfeigne  de  la//poulle 
graffe,  deuant  le  college  de  Cambray.//  i5  56.//Avec  Privilege.' 
(F.  94.) 

The  title  page  of  Book  III  reads  as  follows  : 

'  Le  //  Troysieme  Li-//vre  de  1' Arithmetiqve  //  de  P.  Forca- 
del  de  Beziers.// Avqvel  sont  traictees  les  de-//monftrations 
de  toutes  fortes  de  racines,  auecques  I'entiere  pra-//ctique  de 
I'extraction  d'icelles,  enfemble  plufieurs  queftions,  reigles,// & 
demonftrations  Mathematiques,  auecques  le  propre  fubiect  de 
I'Algebre.// Le  tout  de  I'inuention  de  rAutheur.//A  Paris,// 
Chez  Guillaume  Cauellat,  a  I'enfeigne  de  la  //  Poulle  graffe,  de- 
uant le  college  de  Cambray.//  1 5  5 7-//  Avec  Privilege.'  (F.  i ,  r., 
of  Part  III.) 

Description.  4°,  13.2  X  19.2  cm.,  the  text  being  10.  i  X  16  cm. 
323  ff.  (5+93,  no,  4+  III,  in  the  above  three  books);  25-31  11. 
Paris  1556-57. 

Editions.    Forcadel  wrote  four  arithmetics,  as  follows  : 

1.  '  L'arithmetiqve,'  Paris,  1556-57,  here  described. 

2.  '  L'Arithmetique  par  les  gects,'  Paris,  1558,  8°. 

3.  'Arithmetiqve  entiere  et  abregee,'  Paris,  1565  (p.  316). 

4.  '  Arithmetique  demonftree,'  Paris,  1570,4°. 

This  is  perhaps  the  most  elaborate  French  treatise  on  arithmetic 
published  in  the  sixteenth  century.  Its  three  books  of  about  one  hun- 
dred pages  each  form  a  work  of  the  nature  of  the  great  contemporary 
Italian  arithmetic  of  Tartaglia,  or  rather  of  Tonstall's  Latin  treatise.    Of 


PRINTED   BOOKS 


285 


L'ARITHMETICQVE   DE   P. 

Forcadel  deBeziers.     /  /  ^s£  ■  y*»r  ^  ^' 

VeU  co^oijfance  (^  dljpofitiondes  figures  ,autrement  des 
ftombresJimpUs. 


o  VT  E  rArithmcticquc  confiftca  fjauoir  cognoi- 
ftrcdix  figures  ,  ou  caraderes ,  proprcs  guidons  de 
route  computation  Mathemaiicqucqui  font  cy  dcf 


fousrcngez  felon  leurpropre  nature. 


•UAl.    1.    3. 

•xnapz*  3.  4 

•sioijj.  4.  f. 

•5aatiil)4.  5.  6. 

•bup  5.  d.  7, 

•xijtf.  7.  8 

•adaj  7.  8.  $ 

•amij  8.  p»  0. 

•arg^jnaa^.o.  1. 

•a^nunoojazo.  i.  2. 


c 

fS  cr  .  ^.•^'^  o 

cr  G  vs  tJi  ^   G  N 

4. 5.  6.  7.  8.  p.  o.  zero. 

5.  6,  7.  8.  5).  o.  I.    premieres, 

<>.  7".  8.  5>,  o.  x.,i.  fecondcs, 

7.  S.  51.  o.  I.  1.  5.    tierces. 

8.  p.  o.  I.  2.  3.  4.  quartes. 

9.  o.  I.  2.  5.  4.  5.     quintes. 

0.  I.  2.  5.  4.  5.  6.  fixiefines. 

1.  2.  5.  4. 5.  6, 7.  feptiefmes. 

2.  3.  4. 5.  6.  7.  8.  huitiefmes. 
5.  4. 5.  6.  7.  8.  s).  ncufiefincs. 
e.^  2.  »  ?r>  ar  3 

^  g  3  2.-^  £.  S 
2  5  a*  3^  ^  2.  ?* 


R? 


Fig.  139.    First  page  of  text  of  the   1556-57   forcadel 


286  RARA   ARITHMETICA 

practical  problems  it  has  very  few,  and  it  is  equally  deficient  in  the 
theories  of  the  ancient  arithmeticians.  It  is  simply  a  ponderous  work 
on  the  theory  of  arithmetical  calculations  and  rules,  valuable  for  a 
scholar  but  useless  as  a  practical  textbook. 

For  a  biographical  study  of  Forcadel  see  Boncompagni's  Biillctiiio, 
II,  424. 

Other  ivorks  of  1356.  Baeza,  p.  269,  1555  ;  Cathalan  (Anonymous), 
p.  269,  1555  ;  Gemma,  p.  200,  1540  ;  Medlerus,  p.  223,  1543  ;  Medicus, 
p.  290,  1557  ;  Psellus,  p.  168,  1532  ;  Recorde,  p.  214,  c.  1542  ;  Riese, 
p.  139,  1522  ;  Ringelbergius,  p.  168,  1531  ;  Xylander,  p.  356,  1577  ; 
Anonymous,  '  Arithmetices  Epitome,'  Freiburg,  12°;  A.  Lottini, '  Calculi 
e  conti  per  queUi  che  hanno  denari,'  Lyons,  8°  ;  Casp.  Pauerus,  '  Logis- 
tice  astronomica,'  Wittenberg,  8°  ;  Joannes  Pierius  Valerianus,  '  Hiero- 
glyphica  sive  de  Sacris  Aegyptiorvm  Uteris  Commentarii'  (ancient  finger 
reckoning),  Basel,  fob,  with  editions  ib,,  1567,  fol, ;  ib.,  1575,  fol.;  Lug- 
duni,i579,fol. ;  Juan  Diaz  Freyle,  'Sumariocompendioso  . . .  conalgunas 
reglas  tocantes  al  Aritmetica,'  Mexico  (the  first  arithmetic  printed  in 
America). 

ROBERT   RECORDE.     Ed.  pr.  1557.  London,  1557. 

See  p.  213. 

Title.    See  Fig.  140. 

Colophon.  'Cimprinted  at  London, //by  Jhon  Kyngfton.// 
Anno  domini.     1557.'    (F.  164,  v.) 

Description.  4°,  13  X  18  cm.,  the  text  being  8.1  X  14.7  cm. 
164  ff.  imnumb.,  36  11.     London,  1557. 

Editions.    There  was  no  other  edition. 

Recorde  speaks  of  this  work  as  '  The  feconde  parte  of  Arithmetike, 
containyng  the  extraction  of  Rootes  in  diuerfe  kindes,  with  the  Arte  of 
Coffike  nombers,  and  of  Surdes  nombers  also,  in  fondrie  fortes.'  It  is 
not,  however,  purely  algebraic,  for  the  first  half  of  the  book  is  a  treatise 
on  Boethian  arithmetic.  For  example,  the  following  is  one  of  the  defi- 
nitions :  '  A  Diametralle  nomber,  is  fuche  a  number  as  hath  twoo  partes 
of  that  nature  :  that  if  thei  bee  multiplied  together,  thei  will  make  the 
faied  diametralle  nomber.'  The  treatise  on  '  Cossike  nombers  '  begins  on 
f .  S  I ,  and  *  The  rule  of  equation,  commonly  called  Algebers  Rule'  on 
f .  Gg  4.    The  sign  of  equality,  '  a  paire  of  paralleles,  or  Gemowe  lines  of 

one  lengthe,  thus  : ,  bicaufe  noe  .2.  thynges,  can  be  moare  equalle,' 

is  found  for  the  first  time  in  print  on  f.  Ff  i  (see  Fig.  141). 


PRINTED    BOOKS  287 


oftoftte, 

tofticfte  te  tfft  feconDe  parte  of 

Arithmctiketcontnfnpng  tl^mrac-^ 

t(on  of  IJootcs:  fiCbc  CoJ^i^e  p;aaife, 

loftb  tht  ruU  of  %M4^on:ano 

t^ett)oo;te0of5We 


Tboiigb  manyfiones  doe  iearegreate  price. 
The  Tvhetttone  is  for  exerfice 
A%  neadeful\,ani'm  "booths  asfiraunge: 
0ulle  tbinges  and  harde  it  "ffillfo  cbaange, 
jfndmakf  tbem/harpe,to  right  good  vfc: 
Ml  artefinen  Ipofi/eytbei  can  mt  cbufe, 
Sut  i^ft  b(£  helpe.ytt  as  mm  fee, 
Koep>arf>€nejffefmeth  in  it  to  bee. 

TAf  groundeofartes  didhr«de  tbajlone- 
His  v/i  isgreate/mdmoare  tben  one. 
Here  if^  ou  liflyour%ittes  to  'tDhette, 
Mocbejharpenejfe  therbjfbally  ougette. 
J)ulle  Utittesbereby  doegredtely  mtnde, 
Sbarpe  "^tttes  arefinsdto  their fulle  (vde, 
Kot^proue,andprai/e,asyou  doejinde, 
jtndtoyourfel/Le  notvnkjnde. 


GCIiefe  ^ooKeis  are  to  bee  folDe,at 

rbe  Melt$(oo;e  of  )aouU5« 

i>?3i^onfe?ngaone. 

Fig.  140.   Title  page  of  recorde's    Whctjtonc  of  wittc 


288  KARA   ARITHMETICA 

Theyfrte 
as  tljcic  luo^fef  s  Doe  cptcntjc )  to  oiamctc  it  oitclp  fttto 

tlDOOpartCf.  2JIl6crcoftI}Cfiiaei0,  t»henonenomberis 
equtllcvnto  one  other.  anOt^C  kconnti^.tvbcnonenomf 
itrii  compared  as  e<imlley>nto.7t9tbcrnmberi, 

5llU)aic5  toiUgng  ?on  to  rcmrfacr,  tl)at  pou  rcDucc 
vournombers ,  tottjetrleaOc  ocnominattons ,  and 
fmallette  fo?meiJ,befo?e  poti  pioceuc  anp  farther. 

anD  agaiii,if  pour  r^tt4f/fl>!  be  fofbc,  tl)at  tlje  grea* 
telle  ocnommatton  0/?%,  be  t'oineo  to  anp  parte  of  a^ 
compounuc  nombcr ,  pou  fl^all  tournc  it  To ,  tbat  tbe 
nombcroftl)egreateac  Cgne alone,  maicftanDeas 
cqimllctotberefte. 

ano  tbts  is  all  tbat  neaDctb  to  be  taugljte ,  conce^- 
npngtbisluoojUe. 

^^otubeitjfoj  eaCc  alteratio  tXe^aafmi.^  M\\  p^o^ 
potinDe  a  feloe  eraples^bf  caufe  tbe  ertrattton  of  ti^etc 
rootc0,maietbe  mo;tc  aptlp  bee  tojougbte.  ano  to  a^ 
uoiDetbetebioufe  repetition  of  tbefe  luoo^Oes :  ise# 
qualle  to :  3J  luill  fette  as  31  Hoe  often  m  U)oo;rfec1jre,a 
paireofparalleles,o;(II5cmoiDc  lines  of  one  lengtbe, 
ti)us:=...=~,bicaurc  noe.2.  tbpnges,can  be  moare 
equalle.  ano  nolo  marUe  tbefe  nombers. 


14 

•^•- 

_|— .iy.^==^= 

--7 1.?. 

2o 

1 

'•Sr 

1 

4.     1 9.5^—- (— 1 92.^=— I  o^ — H-io8f 19^ 

6.      545^ I2S^-=-4o::e.~"^4^°"y ^'5" 

I.         in  tl)c  firftc  tbcrc  nppearctfj.  i ,  nombe  rs ,  tbat  is 

1 4.i^ . 

Fin.  141.    From  recorde's    Whctj'tonc  of  witte 


PRINTKI)    H0(;KS  289 

LUCAS  LOSSIUS.    Ed.  pr.  1557.     Frankfort  a.  d.  Oder,  1557. 

A  German  arithmetician  of  the  sixteenth  century,  born  at  Liineberg. 

Title.    See  Fig.  142. 


ARITHME^ 

TICES  EROTE 

MAT  A  PVE- 

R  I  L  I  A. 


IN  QVIBVS  SEX  SPECIES 

huius  vtilifsimcS  arti's ,  ^  Regula  >  quam 

vocant,  Detri,  breuiter  Si  per- 

rpicue  traduntun 

IN  GRATIAM  ET  VSVM  SCHOLA= 

r\m  puerilitm  Ldtmrum  coUe^la ,  cr  iti 

hem  im  recais  edita^ 

A 

LFCA  LOSSIO 

Lunehurgenji. 


FRANCOFORDI7E    AD    ODERAM 

IN  OFFICINA  lOHANNIS 

EICHORNL 

Fig.    142.     I'lTLE  PACE    OF    LO.SSIUS 

Description.  8°,  9.2  X  14  cm.,  the  text  being  6.8  x  11.9  cm. 
4  ff.  unnumb.  +  27  numb.  =  31  ff.,  22-24  l^-  Frankfort.  For 
date  see  f.  4,  v.:   '  Datoe  Lunebar,  Anno   1557. //7-  Februarij.' 


290  RARA   ARITHMETICA 

Editions.  Frankfort  an  der  Oder,  1557,  8°  (here  described); 
s.  1.,  1562,  8°;  Leipzig,  1568,  8°;  Frankfort,  1569;  Magdeburg, 
1585,  8°. 

A  small  book,  intended,  as  the  title  states,  for  beginners  in  the  Latin 
schools.  It  is  arranged  on  the  catechism  plan,  a  feature  not  common 
with  arithmetics  printed  in  Germany  at  this  time,  although  extensively 
used  by  Recorde  in  England  (see  p.  210).  It  begins  :  '  Quid  eft  Arith- 
metica?  Est  bene  &  artificiofe  numerandi  &  computandi  fcientia.' 
(F.  5.)  The  '  species '  are  treated  in  a  practical  way,  but  the  problems 
are  all  traditional,  part  of  them  being  based  upon  biblical  incidents. 

Other  works  of  1557.  Archimedes,  p.  228,  1544;  Camerarius, 
p.  263,  1554;  Gemma,  p.  200,  1540;  Gutierrez  de  Gualda,  p.  167, 
1531  ;  Jacob,  p.  298,  1560;  Paxi,  p.  80,  1503;  Psellus,  p.  170,  1532  ; 
Ramus,  p.  263,  1555  ;  Rudolff,  p.  153,  1526  ;  Tagliente,  p.  115,  15 15  ; 
Martinus  Carolus  Cressfelt,  '  Arithmetica,  Reeckeninge  op  den  Linien 
end  Cyfferen  na  allerley  Hantieringe,'  Deventer  (second  edition  1577)  ; 
Sixtus  Medicus,  '  De  Latinis  numerorum  notis,'  Venice,  4°  (colophon 
date,  1556;  hardly  an  arithmetic). 

GIOVANNI  FRANCESCO  PEVERONE. 

Ed.  pr.  1558.  Lyons,  1581. 

An   Italian   arithmetician   of   the  sixteenth  century,  bom   at   Cuneo,  in 
Piedmont. 

Title.    See  Fig.  143. 

Description.  4°,  15.3  X  21.7  cm.,  the  text  being  10.2  X  17.6 
cm.  136  pp.  (60  on  arithmetic),  35-37  11.  The  dedicatory 
epistle  is  dated  1556.  Bound  with  this  is  'II  breve  Trattato// 
di  Geometria.'     Lyons,  1581. 

Editions.  Although  the  dedicatory  epistle  is  dated  1556,  and 
the  portrait  of  the  author  bears  the  date  1550,  I  know  of  no 
edition  before  1558,  Lyons,  8°.  This  edition  of  1581  seems  to 
have  been  the  second. 

The  first  part  of  the  work  treats  of  the  operations  with  integers.  It 
is  not  a  particularly  progressive  textbook,  as  is  seen  in  the  fact  that  it 
includes  the  '  gelosia'  multiplication  of  the  early  Venetian  writers.  The 
second  book  relates  to  fractions,  the  third  to  business  operations,  and 
the  fourth  to  roots.    The  work  is  in  no  sense  a  scholarly  production. 


Fig.  143.   Title  paoe  ok  peverone 


292  RARA   ARITHMETICA 

ALVISE  CASANOVA.     Ed.  pr.  1558.  Venice,  1558. 

A  Venetian  teacher  of  the  sixteenth  century. 

Title.    See  Fig.  144. 

Description.  4°,  15.5  X  20.8  cm.,  the  text  being  12.5  X  16. i 
cm.     136  ff.  (partly  numbered),  33-38  11.     Venice,  1558. 

Editions.    There  was  no  other  edition. 

Although  this  rare  work  is  usually  classed  as  a  commercial  arithmetic, 
it  is  rather  a  collection  of  bookkeeping  problems.  As  such  it  was  one 
of  the  most  prominent  of  the  century,  and  it  gives  an  idea  of  the  busi- 
ness questions  of  the  Venetian  merchants  of  its  time. 

Other  works  0/1358.  Anianus,  p.  32,  1488;  Archimedes,  p.  228, 
1544  ;  Forcadel,  p.  284,  1556-57  ;  Gemma,  p.  204,  1540  ;  Glareanus, 
p.  192,  1539  ;  Mariani,  p.  181,  1535  ;  Medlerus,  p.  223,  1543  ;  Recorde, 
p.  213,  c.  1542;  Riese,  p.  141,  1522;  Andreas  Clatovenns,  'Arith- 
metica  Bohemice,'  Prag,  8°  (Wydra,  in  his  Historia  Matheseos  in 
Bohemia,  p.  18,  mentions  it  as  printed  in  Niirnberg,  1530).  There  were 
also  two  arithmetics  published  s.  a.,  but  c.  1558,  viz. :  Benese,  p.  182, 
c.  1536  ;  Gemma,  p.  200,  1540. 

JOHANNES  BUTEO.     Ed.  pr.  1559.  Lyons,  1559. 

BoTEO,  BuTEON,  Bateon.  Bom  in  Dauphine,  c.  1485-1489;  died  in  a 
cloister  in  1560  or  1564.  He  belonged  to  the  order  of  St.  Anthony,  and 
wrote  chiefly  on  geometry,  exposing  the  pretenses  of  Finaeus. 

Title.    See  Fig.  145. 

Description.  8°,  10. 4  X  16.9  cm.,  the  text  being  7.2  X  13.2  cm. 
400  pp.,  23-27  11.     Lyons,  1559. 

Editions.  Lyons,  1559,  8°  (here  described);  ib.,  1560,8°.  An 
edition  of  his  'Opera'  appeared  in  1554. 

The  work  is  divided  into  five  books,  the  first  treating  of  the  funda- 
mental operations  with  integers,  the  second  of  fractions  and  the  rule  of 
position,  the  third  of  algebra,  the  fourth  of  arithmetical  problems,  and 
the  fifth  of  algebraic  problems.  The  problems  are  not  of  practical 
value,  and  hence  the  arithmetic  never  attained  any  popularity. 

Other  7Vorks  0/1359.  Albert,  p.  180,  1534  ;  Anianus,  p.  32,  1488; 
Cataneo,  p.  244,  1546  ;  Gemma,  p.  200,  1540  ;  Mariani,  p.  181,  1535  ; 
Johann  Fischer  (Piscator),  p.  247,  1549  ;  Herman  Giilfferich,  '  Ein  new 
Rechenbiichlin  auff  der  Linien  und  Federn'  (perhaps  the  work  men- 
tioned on  p.  257,  1552);  Pedro  Juan  Monzo  (Monzoni),  '  Elementa 
Arithmeticae,'  Valencia,  8°,  and  s.  1.,  1566,  1569. 


PRINTED    BOOKS 


293 


Fig.  144.    Title  page  of  casanova 


294 


RARA   ARITHMETICA 


B  V  T  E  O  N  IS 

LOGISTIC  A,   ^  A 

&  Arichmecicavulgo  dicicur  in  Ii- 

bros  quinque  digefta:qiio- 

rum  index  fummatim 

habetur  in  tcrgo. 

E  1  r  ST)  E  M, 

^d  locum  VttYHuij  corrHptHm  ■^'esHtutioyi^ui  ejl 

de  proportioKe  Upidum  mittendorumadbalifls 

foramefi^Lihro  Decimot 


L  V  G  D  V  N  I» 

JifW    CrLlELMVM   ROVILLIVM^ 

5VB    SCVTO    VENETO. 


J/i.   D.   LIX. 

Cum  priuilegio  JRegis, 

Fig.  145.   Title  page  of  buteo 


PRiN'ri:i)  liooKs  295 

FRANCISCUS  BAROCIUS.     Ed.  pr.  1560.         Pavia,  1560. 

Fkancksco  Baruzzi.    Born  at  Venice,  c.  1538  ;  died  after  15S7.    He  wrote 
on  cosmography,  and  edited  Proclus. 

Title.  '  Francisci  Barocii  //  patritii  Veneti  //  opvscvlvm,//  in 
quo  vnaOratio,&cluacQueftionc.s:// altera  de  certitudine,&  altera 
//de  medietate // Mathematicarum  continentur.// Ad  Reueren- 
disfimum  Danielcm  Barbarum  Patriarcham//Aquileienfem  defiji^- 
natum  Virum  Clariff.// Patavii,  E.  G.  P.//M.  D.  LX.'    (F.  i,  r.) 

Description.  4°,  14.  i  X  19.6  cm.,  the  text  being  10.  i  x  15.3 
cm.     40  ff.  numb.,  26  11.     Pavia,  1560. 

Editions.    There  was  no  other  edition. 

This  philosophical  discussion  has  been  mentioned  because  it  includes 
some  reference  to  the  old  Boethian  arithmetic.  No  arithmetical  pro- 
cesses are  discussed. 

OLIVIERO  FONDULI.     Ed.  pr.  1560.  Bologna,  1560. 

A  Bolognese  teacher  of  the  middle  of  the  sixteenth  century. 

Title.    See  Fig.  146. 

Description.  8°,  9.9  X  14.6  cm.,  the  text  being  6.9  x  i  i.i  cm. 
23  ff.  unnumb.  (possibly  one  missing  at  the  end),  24  11.  The 
date  is  given  on  f .  i,  v.:  'Dat.  Bonoiae  .  die  xxv.  Septemb.  1560.' 
Bologna,  1560. 

Editions.    There  was  no  other  edition. 

This  rare  little  handbook  of  commercial  customs,  from  the  press  of 
Pellegrino  Bonardo  of  Bologna,  is  hardly  an  arithmetic,  although  it 
explains  certain  arithmetical  processes,  and  gives  a  considerable  number 
of  business  problems. 

SIMON  JACOB.     Ed.  pr.  1560.  Frankfort,  1565. 

Born  at  Coburg;  died  at  Frankfort  am  Main,  June  24,  1564.    He  was  one 
of  the  best-known  Rechenmeisters  of  his  time. 

Title.    See  Fig.  147. 

Colophon.  '  Getruckt  zu  Franck-//furt  am  Main/  bey  Georg 
Raben///in  verlegung  Sigmund  F"eyerabends/ //vnd  Simon 
Huters.  Sigmund  P^eyerabent // Simon  Hutter//M.  D.  LXV.' 
(F.  363,  V.) 


296 


KARA   ARITHMETICA 


PRATICHE 

DE   FIORETTI   MERCHANTILI. 

vtilifsimeaciafcheduna  perfbna ,  di  mandare  a 

memorialebreuelnuetioni  fabrichate  fbpi* 

jl  Valutar  de  pefi ,  &  mifure,  &  altre  in 

ftruttioni  necceflarie  da  fapere . 

Ec  anchora  a  quadrare  Muraglie ,  Taffclli ,  &  Coperti  »■ 

Ttude ,  Fieno  ,  &  Legne.-Con  la  decchiaratrone,  & 

Exempli  loro  come  legendo  intenderai. 


Ficj.  146.   Title  page  of  fonduli 


PRINTHU    liOOKS 


297 


extradion  Radicum^viT  toott  hm  ^to\}oitionm/mit  vilcn  f  uf^igcn  Sragcn  vtl 

Q{iiffga6f»/;f.    *^ff5gI^ic^m<mboIIrolrSlKr:5eric^t^fc9vc3cI,'fa[ftmitluutt3ell juuftitio* 

MihJ8/^mo»(l?i(tpnibu8/bn&»ortf><llfn/fo&i|jaitf;crfiir»nmftg[ic^gffc^t(3f/3tb«lttrt/l)fr» 

§I<ic^en noc^  nU an  tag  romme».    «  CDiiC  bann  Bon Det  Ocomctria/  wie  man raojic^etk!;  5<U 

&cr  J>u6  t6nf /auc^  aflerlrt;  (f  ospora/TifguJarJa  biib  3rregufaria/nKjTcn/2(«am  pnbcn  t>fi  r«c^ 

nen  fol,  3illf s  t»urc^  0imen  3aco6  t>on  (ToBurg/  'burger  toiiD  0^ec^enm«i(l(r  iu 

5r«ntf  fiitt  am  Wain/mit  f[ci|j  jufanimen  getragen  /  bnb  1  egt 

er(lmal0g«r»cff. 


(Betrucf  t  ju  ^ranctfurr  am  W7*yn/ » ;  «  /♦ 

Fig.  147.   Title  page  ok  jacob 


298  RARA   ARITHMETICA 

Description.  4°,  15.2  X  18.6  cm.,  the  text  being  10.3  X  15  cm. 
354  ff.  (349  numb.),  30  11.     Frankfort,  1565. 

Editions.    Jacob  published  two  works,  as  follows  : 

1.  '  Rechenbuchlein  auf  den  Linien  und  mit  Ziffern,'  Frank- 
fort, 1557,  12°;    1574,  12°;    1589;    1590;    1599- 

2.  '  Kin  new  vnd  wolgegrundt  Rechenbuch,'  1 560  ;  Frankfort, 
1565,  4°  (here  described);  1569;  1600,  4°  (below),  and  in  the 
seventeenth  century. 

Jacob's  arithmetics  followed  the  general  plan  of  the  popular  books 
of  Riese,  and  were  deservedly  well  received  in  the  second  half  of  the 
century.  They  were  commercial  textbooks,  and  although  they  do  not 
show  any  mathematical  advance  they  are  historically  valuable  for  their 
applied  problems.  The  title  page  (fig.  147)  is  interesting  because  of  the 
variety  of  mathematical  instruments  illustrated. 

Other  works  of  1560.  Borghi,  p.  16,  1484  ;  Buteo,  p.  292,  1559; 
Feliciano,  p.  149,  1526  ;  Psellus,  p.  168,  1532  ;  Scheubel,  p.  246,  1549  ; 
Tartaglia,  p.  278,  1556  ;  Gasparo  Rizzo,  '  Abbaco  nvovo  molto  copioso 
et  artificiosamente  ordinate,'  Venice,  8°  ;  Juan  Ventallol,  '  Aritm^tica,' 
a  rare  work,  s.  1.  a.,  which  appeared  in  Spain  about  this  time. 

SIMON  JACOB.     Ed.  pr.  1560.  Frankfort,  1600. 

See  p.  295. 

Title.  This  is  practically  the  same  as  in  the  1565  edition 
described  above. 

Colophon.  '  Getruckt  zu  Franckfurt  am  Mayn/  bey  //  Matthes 
Beckern/  In  Verlegung  Chris-//ftian  Egenolphs  Erben.//  Anno 
1600.'     (F.  360,  V.) 

Description.  4°,  15.5  X  19.2  cm.,  the  text  being  10.5  X  14.8 
cm.    1 1  ff.  unnumb.+  349  numb.  =  360  ff.,  28  11.   Frankfort,  1600. 

See  above. 

JOSEPHUS  UNICORNUS.     Ed.  pr.  1561.         Venice,  1561. 

Giuseppe  Unicorno.    A  Bergomese  arithmetician,  born  in  1523;  died  in 
1610. 

Title.    See  Fig.  148. 

Description.  8°,  9.4  X  13.9  cm.,  the  text  being  7.4  X  12.7  cm. 
I  f.  unnumb.  +  78  numb.  =  79  ff.,  30  H-    Venice,  1561. 


PRINTED    HOOKS 


299 


lOSEPHI   VNICORNI 

BERGOMATIS 
LIBER 

DE    VTILITATE  MATHEMATICA* 
R    V.   M       A   K    T    I    V    M, 

In  quocitniidcLe^ormulta  quidem  fckuperiucun^ 
dd ,  admiratione  digna,  CT  humano  ufid  ' 
mceffitrU  pafsim  repcries. 


V  B  N  B  T  1 1  s ,  Apud  Dominicum  de  NicoUnis , 
Anno  Domini,     M    D     L  X  I. 

Fk;.   i4<S.  Title  page  of  unicornus 


300  RARA   ARITHMETICA 

Editions.    There  was  no  other  edition. 

This  work  is  not  an  arithmetic,  Unicorn's  textbook  on  this  subject 
appearing  much  later  (p.  412,  1598).  It  is  a  proHx  dissertation  on  the 
uses  of  mathematics,  and  of  arithmetic  in  particular,  with  extracts  from 
the  ancient  and  mediaeval  writers.  Like  Agrippa's  '  De  vanitate  scien- 
tiarum,'  it  is  interesting  but  profitless. 

NICOLAUS  WERNER.     Ed.  pr.  1561.  Nurnberg,  1561. 

A  Nurnberg  Rechenmeister,  born  c.  1520. 

Title.    See  Fig.  149. 

Colophon.  'Gedriickt  zu  Nurnberg/  //durch  Johann vom  Berg/ 
//vnd  Ulrich  Newber.'     (F.  156,  r.) 

Description.  4°,  14.2  X  19.3  cm.,  the  text  being  10. i  X  15.5 
cm.     156  ff.  unnumb.,  23-31  11.     Nurnberg,  1561. 

Editions.    There  was  no  other  edition. 

This  is  a  purely  mercantile  arithmetic,  and  like  many  of  the  Italian 
textbooks  it  assumes  some  preliminary  knowledge  of  the  fundamental 
operations.  It  begins  with  a  treatment  of  Welsch  (Italian)  practice, 
and  is  composed  almost  entirely  of  practical  problems  of  the  day,  solved 
by  this  method.  Welsch  practice  differs  from  the  rule  of  three  only  in 
having,  ordinarily,  unity  for  the  first  term,  as  in  the  following  example  : 
If  I  book  cost  $2,  how  much  will  7  books  cost?  This  method  was  very 
popular  with  all  German  arithmeticians  of  the  sixteenth  century,  and 
survived  in  the  chapter  on  Practice  still  to  be  found  in  English  arith- 
metics. It  appeared  also  in  the  early  American  textbooks.  The  book 
gives  a  very  good  idea  of  the  business  requirements  of  the  second  half 
of  the  sixteenth  century.  The  value  placed  upon  exchange  and  '  profit 
and  loss '  may  be  seen  in  the  fact  that  twenty-six  pages  are  devoted  to 
the  former  and  forty-seven  to  the  latter,  while  partnership  also  has 
twenty-six  pages  for  its  share. 

JOHANNES  MONHEMIUS. 

Ed.  pr.  1561.  Diisseldorf,  1561. 

A  German  teacher  of  the  second  half  of  the  sixteenth  centui7. 

Title.    See  Fig.  i  50. 

Description.    8°,   9.4  X  15   cm.,  the  text  being  6.5  x  12  cm. 
22  ff.  unnumb.,  25-27  11.     Diisseldorf,  1561. 
Editions.    There  was  no  other  edition. 


PRINTED    BOOKS 


30  ] 


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P'iG.  149.   Title 


PAGE    OF    WEKNER 


302  RARA   ARITHMETICA 

This  work  is  almost  unknown  to  students  of  arithmetic.  It  seems  to 
be  the  only  edition,  although  the  introduction  bears  the  date  1542.  It 
is  a  small  work  and  is  based  somewhat  on  the  mediaeval  texts.    It  gives 

M  E  T  H  O- 

VrS  J^ITHMETICES 

COMPVTATORlAE,OMNHM  SVP^ 

pMUndi  dYtem  trddcni:tmHUm,qn£mU% 

nmeroYum.qum  qu£  oUm  cakuUs, 

nunc  nummisfuper  £que  di^M 

tesUneas  furfum  ac  d€t 

crfum  pofiiis, 

pcrficitur* 

AuSlorelo.  Monhemio^ 


.^^''Si^^'JlIl^^^s^ 


DVSSELDORPH 
Anno  I  $  ^  ^' 

Fig.  150.   TrrLE  page  of  monhemius 

the  fundamental  operations  with  integers  and  a  brief  treatment  of  the 
rules  of  three,  partnership,  and  false.  There  is  also  a  brief  explanation 
of  counter  reckoning. 


PRINTED    HOOKS  303 

BENEDICTUS  HERBESTUS.    Ed.  pr.  i  561.    Craonv,  1577. 

Born  at  Novomiasti,  Toland,  in  1531  ;  died  at  Jaroslaw,  March  4,  1593. 
He  was  a  Jesuit  priest  at  Cracow. 

Title.    See  Fig.  151. 

Description.  8°,  9.5  X  15.3  cm.,  the  text  bcinj;  8  X  11.5  cm. 
42  ff.  unnumb.,  27-28  11.     Cracow,  1577. 

Editions.  1561;  Cracow,  1564,8°;  ib.,  1566;  ib.,  1569;  ib., 
1577,  8°  (here  described). 

A  Latin  book  intended  for  the  Church  schools,  and  revised,  as  the 
'  Prosemivm  auctoris  '  states,  '  Mariaiburgi  in  Prufsia,  fub  finem  Anni 
Domini  1576.'  It  consists  of  two  parts,  the  first  being  devoted  to  the 
arithmetical  operations,  chiefly  with  counters  (and  this  as  late  as  1577), 
and  the  second  to  the  calendar.  Besides  the  operations,  the  first  part 
also  includes  a  chapter  on  progressions  ('  De  Progressione.  Capvt  VII.'), 
and  one  on  the  rule  of  three  ('  De  tribvs  nvmeris  integris.    Capvt  VIII.'). 

ANDREAS  HELMREICH.     Ed.  pr.  1561.     Eisleben,  1561. 

A  Halle  Rechenmeister  of  the  latter  half  of  the  sixteenth  century.  The 
dedication  of  the  1561  edition  is  signed  by  '  Andreas  Helmreich  Rechen- 
meifter  vnd  Vifierer  zu  Halle,'  and  the  1588  edition  by  '  Andreas  Helmreich 
von  Eifzfeldt/  Notarius  publicus/  Rechenmeifter  vnd  Vifierer  zu  Halle.' 

Title.    See  Fig.  152. 

Colophon.  'Gedruckt  ai  Eisleben/ bey  yV  Vrban  Ganbifch.' 
(F.  Ill,  r.) 

Description.  4°,  14.2  X  19.3  cm.,  the  text  being  9.3  X  14.9  cm. 
Ill  ff.  unnumb.  +  i  blank  =  112  ff.,  27-33  II.    Eisleben,  1561. 

Editions.  This  is  probably  the  first  edition  of  Helmreich's  arith- 
metic, the  dedicatory  epistle  being  dated  'zu  Halle  inSachffen  den 
Sontag  Oculi/  welcher  war  der  9.  tag  des  Monats  Marcij/  nach 
Chrifti  Ihefu  vnfers  lieben  Herrn  vnd  Seligmachers  Geburt  1 56 1 .' 
Murhard  (1, 155)  mentions,  however,  an  edition  appearing  at  Halle 
in  1546,  4°.  The  book  was  republished  at  Leipzig  in  1588  (p. 
306),  and  again  in  1595,  4°  (p.  306),  and  possibly  ib.,  1596,  4°. 

The  work  is  an  unsuccessful  attempt  to  combine  the  old  and  new 
arithmetics.  The  author  begins  with  a  semi-Boethian  treatment  of 
ratios,  assuming  a  knowledge  of  the  fundamental  operations  with 
integers,  and  then  introduces  a  course  in  mercantile  arithmetic.  He 
closes  with  a  considerable  amount  of  work  on  mensuration,  including 


304  RARA   ARITHMETICA 

ARITHMP 

TICA  LINEA'' 
RIS 

FIGVRATA, 

cum  quibufdam  ex 

COMPVTO 

neceflfarijs; 

auHore:BEtA(^EVICTO 

HEBJSeJloy  Societatis 

lESV  Trejbytero, 

cumfacultate  Superiorum. 

CVM  GRATIA  ET   PRlw7fgo  S,  R.  M. 

Jn  Oficina  Mattbai  Sicbcncy* 

cbcr :  i^nno  DOMitU 

^577. 


Fig.  151.   Title  page  of  the  1577  herbestus 


PRINTED    BOOKS 


305 


toon  tjortcil  m^  fe^cttt^ig 

cracntWc^en/  MiitWc&cn  rn&  pnterfc^eDWc^en  proportjen/ 

fampt0ea?(flcr  pn&  bet)fn&er  art/manc^erley  tXHet^ 

rutl[?cn  pn^  0c^nilrc/auff  alle  Ol?me  pnO  jCicb 

5a  mat l?en/5uramen  bracbtfeurcb 


Fig.  is 2.   Title  page  of  the  1561   helmreich 


A'ean- 


306  KARA   ARITHMETICA 

gauging  (Visierrechnung).    It  is  somewhat  surprising  that  such  a  wi 
some  treatise  ever  went  through  three  editions. 

Other  works  0/1561.  Albert,  p.  178,  1534;  Borghi,  p.  16,  1484; 
Feliciano,  p.  148,  1526;  Gemma,  p.  204,  1540;  Recorde,  p.  214, 
c.  1542  ;  Rudolff,  p.  152,  1526  ;  Sfortunati,  p.  177,  i534  ;  Tagliente, 
p.  115,  1515;  Wolphius,  p.  154,  1527;  A.  CitoUni,  * Tipocosmia,' 
Venice,  8°  (a  synopsis  of  science,  incUiding  arithmetic)  ;  Ognibene  de 
Castellano,  '  Opera,'  Venice,  4°  (principally  on  geometry,  but  including 
some  arithmetic  ;  see  also  p.  375,  1582)  ;  Antonio  Maria  Venusti, '  Com- 
pendio  utilissimo,'  Milan,  8°  (includes  some  treatment  of  exchange). 

ANDREAS  HELMREICH.     Ed.  pr.  1561.       Leipzig,  1588. 

See  p.  303. 

Title.  '  Rechenbuch// Von  vortheil  vnd  behendig-//keit/  nach 
der  Welfchen  Practica/  mit  //  jhren  vnterfchiedlichen  Propor- 
tionen.//Grund  vnd  vrfach  der  Regel  Detri/  // Sampt  gewiffer 
vnd  behender  art/  mancherley  Vi-//fier  Ruthen  vnd  Schnure/ 
nach  dem  Quadrat  vnd  //  Cubo,  auff  alle  Ohme  vnd  Eiche  zu 
machen/  alles  mit  war-//hafftigen  perfpectiuifchen/  Geometri- 
fchen  vnd  Arithmetifchen  demon-//ftrationen,  aufz  den  Buchern 
Euclidis  fundirt/mit  fleifz //zufammen  gebracht  vnd  in  Druck 
aufzgangen///Durch//Andream  Helmreich/Rechenmeifter  vnd 
Vifierer  //  zu  Hall  in  Sachffen.//  (Cut  showing  dimensions  of  a 
cask.)    Im  Jahr  1588.'     (P.  3.) 

Colophon.  'Gedruckt  zu  Leipzig/ //Durch  Abraham  Lamberg 
//Anno//M.  D.  LXXXVIIL'     (R  336.) 

Description.  4°,  15.5  X  19.2  cm.,  the  text  being  10.  i  X  14.5  cm. 
9  pp.  blank  +  i6unnumb.  +  317  numb.  =  342  pp.,  31  11.  Leipzig, 
1588. 

See  p.  303. 

ANDREAS  HELMREICH.     Ed.  pr.  1561.       Leipzig,  1595. 

See  p.  303. 

Title.  '  Rechenbuch/  //  Erftlich/ Von  //  Vortheil  vnnd  Behen- 
//  digkeit/  nach  der  Welfchen  Practica/  mit  ih-//ren  vnterfchied- 
lichen Proportionibus,  Grund  vnd  vr-//fach  der  Regel  Detri  oder 
proportionum.//n.  Von  zubereitung  mancherley  Vifier  Ruthen/ 


PRINTIiD    HOOKS  307 

// vnnd  Schniiren/damit  alle  VahlTcr  vnnd  andcrc  Corpora  nach 
dcm  //Quadrat  vnd  Cubo  auff  alle  Ohme  vnd  Eiche-Vificre  vver- 
den.//III.  Wic  man  kiinftlich  das  Fcld  vnd  Erdreich/auff  man- 
//chcrley  art/mit  gevviffcr  Mcliruthcn  vnd  Schnuren/nach  cines 
jedcn  Landcs//oder  Stadt  gebrauch/ Gcometrifchcr  vnd  Jdio- 
tifchcr  weili/  vnd  was  hierV/innen  dcr  Vntcrfcheid  fey/  recht 
meffen  fol.//  HH.  De  Diftantijs  Locorum,  das  ift/  Wie  man 
wunderbar-//licher  Geographifchcr  vnd  Cofmographifcher/  nach 
Geometrifcher  weife/  durch  //  Arithmeticam,  in  der  gantzen 
Welt/zvveyer  Stedte  oder  Wonung  Diftantiam,  oder//wie  weit 
die  von  einander  gelegen/  nach  ihren  Longitudinibus,  vnd  Lati- 
tudinibus,  //  fol  Rechncn  vnd  finden/  Sampt  einer  Landtaffel 
daraus  zu  //machen  vnd  zu  befchreiben.// V.  Vnnd  wie  man  die 
Funff  Horologia  communia,  oder  ge-//meine  Sonnen  Vhrn/  als 
Horizontale,  Meridionale,  Septentrionale,  Ori-//t^ntale,  vnd  Occi- 
dentale,  auff  einen  Cubum,  aufzwendig/  oder  fonften  an  die  Wen- 
//de/  Mawer  vnd  ebenen/  Vnd  das  Fundamentum  Horologiorum 
abreiffen.  Item/  //der  Sonnen  vnd  des  Monds  am  Himmel  ge- 
fchwinden  Lauff  rechnen.  Vnd  auch  eine  fonderliche  gar  kunft- 
//liche  Sonnen  Vhr/  der  Cylinder  genandt/  darinne  alle  Stunde 
nach  der  Sonnen  fchein  del^  Tages  grund-//lich  zu  erfehen  vnd 
zu  befinden/ machen/ vnd  fampt  andern  mehr  fchonen  Kunften 
vnd  Vbun-//gen/ nutzlich  gebrauchen  fol.// Alles  mit  warhaff- 
tigen  Demonftrationibus  vnd  Figuren/  aus  den  Buchern  Euclidis 
vnd  andern  fundirt.  Durch  //  Andream  Melmreich  von  EiPfeldt/ 
Rechenmeifter  vnd// Vifierer  zu  Halle  in  Sachfen  an  der  Sahle. 
// Cum  Priuilegio.'     (P.   i.) 

Colophon.  'Gcdruckt  zu  Leipzig/durch  Zacha-//i'iam  Berwald. 
//  Im  lahr/  //  M.  D.  XCV.'     (P.  645.) 

Description.  4°,  14.5  X  187  cm.,  the  text  being  10.9  X  15.4 
cm.  17  pp.  unnumb.  +  3  blank +  627  numb.  =647  pp.,  26-34  11. 
Leipzig,  1595. 

Editions.    See  p.  303. 

This  edition  contains  the  Rechenbuch  of  1561,  together  with  much 
additional  matter  as  summarized  on  the  title  page,  viz.  books  III,  IV 


3o8  KARA   ARITHMETICA 

(pp.  306-631).  This  includes  a  considerable  amount  of  work  on  gaug- 
ing, surveying,  cosmography,  and  the  '  Sphere.'  The  first  304  pages 
are  practically  identical  with  the  first  edition,  except  for  the  addition 
of  numerous  woodcuts. 

GUGLIELMO  PAGNINI.     Ed.  pr.  1562.  Lucca,  1562. 

A  Lucca  mathematician  of  the  sixteenth  century. 

Title.  '  Practica//  Mercantile//  Moderna.//  Di  Guglielmo  Pa- 
gini // Lucchefe.//In  Lucca  per  11  Bus-//dragho.    MDLXIL' 

(P-  5-) 

Description.  4°,  14.5  x  20.6  cm.,  the  text  being  9.  i  X  15.3  cm. 
14  pp.  unnumb.  +  161  numb.  =  I75  PP-.  19-28  H.     Lucca,  1562. 

Editions.    There  was  no  other  edition. 

This  is  a  commercial  arithmetic,  the  only  one  I  have  seen  that 
appeared  in  Lucca  in  the  sixteenth  century.  The  author  begins  with 
the  operations  on  denominate  numbers,  but  places  division  immediately 
after  addition.  The  galley  method  of  dividing  is  not  given  at  all,  in 
which  Pagnini  shows  a  more  advanced  spirit  than  most  of  his  contem- 
poraries. There  is  first  the  '  Modo  di  partire  ditti  partiri  piccoli,'  or 
short  division  ;  then  the  '  Modo  di  partir  per  ripiegho,'  '  Modo  di 
partire  per  colonna,'  and  finally  the  '  Modo  di  partire,  a  danda  per 
altro  nome  partire  grande.'  Following  division  is  the  work  in  multipli- 
cation, including  '  regoletto,'  '  biricuocholo,'  '  crocetta,'  '  per  colonna,' 
and  '  per  ripiegho.'  Subtraction  or  '  traction  '  ('  Modo  di  trare,  o  fottrare  ') 
follows  multiplication.  The  author  then  takes  up  fractions  in  the  same 
order.  This  work  in  the  fundamental  operations  is  followed  by  chapters  on 
exchange  and  mercantile  problems,  the  '  Regola  del  tre,'  profit  and  loss 
('  Gvadagni  e  perdite  '),  partnership  ('  Compagnie  '),  barter,  interest  and 
discount  ('  Meriti  e  sconti  simpHci  e  capo  d'anno'),  and  a  further  treat- 
ment of  exchange.  Altogether  the  arithmetic  is  one  of  the  most  interest- 
ing of  the  smaller  books  of  the  time  issued  under  Florentine  influence. 

JUAN  PEREZ  DE  MOYA.    Ed.  pr.  1562.    Salamanca,  1562. 

Born  in  San  Stefano  (Santisteban  del  Puerto),  in  the  Sierra  Morena,  in 
the  first  third  of  the  sixteenth  century.  He  studied  at  Alcala  and  Salamanca. 
He  was  canon  at  Granada  in  the  latter  half  of  the  century. 

Title.    See  Fig.  153. 

Colophon.  '  En  Salamanca.//Por  Mathias  Gaft.// Ano  de  1 562.' 
(P.  765.) 


PRINTED    BOOKS 

ARITHMETICA 

PRACTICA,Y  SPECV- 

latiuadelBachillerluan 
Perez  de  Moya. 


309 


Agora  nucuamente  corregida ,  y  anadidaj 
por el mifmo author muchas  cofas,con 
otros  dos  libros,y  vna  Tabla  muy  copio 
fade  las  cofas  mas  notables  de  todo  lo 
que  en  efte  libro  fe  contiene. 

Va  dirigida  almuy  altoy  muypodero» 
fofenor  don  Carlos  frincipe 
deEfpaiiamefiro 
ftnOY* 

ConttcmdayfYmlegto  ^al 

EN    5ALAMJINCA« 
PorlilathiasGafl* 
I  5  ^  t 

EiU  callajo  I  moco  blancas  clf Uego^ 

Fig.  153.    Title  page  of  the   1562  moya 


3IO  KARA  ARITHMETICA 

Description.  8°,  9.3  X  14.9  cm.,  the  text  being  7  x  12.1  cm. 
811  pp.  (765  mimb.),  29  11.     Salamanca,  1562. 

Editions.  This  is  the  first  edition  of  this  work  that  I  have 
been  able  to  find,  in  spite  of  the  words  '  agora  nueuamente  corre- 
gida '  in  the  title.  It  appeared  again  at  Alcala  in  1573,  and  at 
Madrid  in  1598,  8°,  and  there  were  at  least  thirteen  editions 
between  1609  and  1706.    For  the  1703  edition  see  below. 

Moya  also  published  three  other  works  :  '  Reglas  para  cotar 
sin  pluma  y  de  reduzir  unas  monedas  castellanas  en  otras,'  in 
1563,  4°;  '  Manval  de  contadores,'  Alcala,  1582,  8°,  and  Madrid, 
1589,  8°;  and  '  Tratado  de  matematicas,'  Alcala,  1573,  fol., 
containing  a  section  on  arithmetic  less  complete  than  the  '  Arith- 
metica  practica.' 

This  is  an  elaborate  treatise  of  765  pages  of  text,  covering  the 
ordinary  calculating  by  algorism,  the  use  of  counters,  business  arith- 
metic, the  elements  of  algebra  (of  which  word  he  gives  the  etymological 
meaning,  adding  the  rest  of  the  ancient  title,  '  almucabala '),  practical 
geometry  ('  Trata  algvnas  reglas  de  Geometria  pratica  neceffarias  para 
el  medir  de  las  heredades,'  p.  304),  and  the  calendar.  Besides  all  this, 
Moya  gives  a  large  amount  of  information  concerning  matters  of  his- 
torical interest.  For  example,  he  treats  the  subject  of  notation  very 
fully,  giving  the  Greek,  Roman,  Hebrew,  digital,  and  astrological  systems, 
together  with  a  brief  mention  of  other  systems.  Altogether  it  is  the 
most  noteworthy  book  on  mathematics  published  in  Spain  in  the 
sixteenth  century. 

JUAN  PEREZ  DE  MOYA. 

Ed.  pr.  1562.  Barcelona,  1703. 

See  p.  30S. 

Title.  '  Arithmetica// Practica,// y  Especvlativa,//del  Ba- 
chiller // Jvan  Perez //de  Moya.//Aora  nvevamente  corregida, 
//y  anadidas  por  el  mifmo  Alitor  mu-//chas  cofas.// Con  otros 
dos  Libros,//y  vna  Tabla  muy  copiofa  de  las  cofas  mas  //nota- 
bles de  todo  lo  que  en  efte  Libro // fe  contiene.// Ano  1703. 
Con  Licencia :  En  Barcelona,  en  la  Imprenta//de  Rafael  Fi- 
guero.'     (F.  i,  r.) 


PRINTED    BOOKS  311 

Description.    8°,  15  X  20  cm.,  the  text  bein<^^  10.7  x  17.9  cm. 
396  pp.  (380  numb.),  39-41  11.     Barcelona,  1703. 
Editions.    See  p.  310. 

See  p.  310.  It  shows  the  popularity  of  this  Spanish  treatise  that  this 
edition  should  appear  141  years  after  the  first  one,  and  that  another 
edition  was  published  as  late  as  1761. 

ANONYMOUS.     Ed.  pr.  1562.  Paris,  1562. 

Title.    See  Fig.  154. 

Description.  8°,  10.5  X  15.9  cm.,  the  text  being  6.9  x  12.5 
cm.    98  pp.  numb.,  24-30  11.    Paris,  1562. 

Editions.    There  was  no  other  edition. 

The  work  is  largely  theoretical,  but  it  contains  a  few  applied  prob- 
lems.   It  was  without  merit. 

Other  works  of  1^62 .  Benese,  p.  182,  1536  ;  Budaeus,  p.  99,  1514  ; 
Euclid,  p.  II,  1482;  Gemma,  p.  200,  1540;  Ghaligai,  p.  132,  152 1  ; 
Lossius,  p.  290,  1557;  Ramus,  p.  330,  1569;  Riese,  p.  139,  1522; 
Xylander,  p.  356,  1577  ;  Francesco  Spinola,  '  De  Intercalandi  ratione 
corrigenda,  &  tabellis  quadratorum  numerorum,  a  Pythagoreis  disposi- 
torum,'  Venice,  8°. 

VICTORINUS  STRIGELIUS. 

Ed.  pr.  1563.  Leipzig,  1563. 

Strigel.  Born  at  Kaufbeurn,  December  26,  1524;  died  at  Heidelberg, 
June  26,  1569.  He  was  professor  of  theology  at  Jena,  Leipzig,  and  Heidel- 
berg, and,  aside  from  the  work  here  mentioned,  wrote  entirely  on  that  subject. 

Title.    See  Fig.  155. 

Description.  8°,  9.  i  X  14.9  cm.,  the  text  being  6.3  X  1 1.8  cm. 
83  ff.  unnumb.  +  i  blank  =  84  ff.,  21-27  ^-     Leipzig,  1563. 

Editions.  There  was  no  other  edition.  It  seems  by  the  pref- 
ace that  the  work  was  written  in  1551.  See  p.  249,  1549.  For 
the  date  of  the  printing  of  this  work,  1563,  see  f.  6,  v.,  of 
the  preface.  F.  10,  v.,  gives  the  year  in  which  it  was  written, 
MDLI. 

This  comparatively  unknown  work  presents  the  subject  of  arithmetic 
from  the  classical  standpoint.  The  author  speaks  of  the  dignity  of 
arithmetic  ('  De  dignitate  arithmeticae  '),  and  follows  this  discussion  by 


312 


RARA   ARITHMETICA 


A  R  I  T  H  M  E- 


T  I  C  A» 


P  A  R.  I  S  I  I  S, 

ApudAndreamWcchelum. 
15^2. 

Cum  privilegio  Regis. 

Fig.  154.   Title  page  of  the  anonymous   1562  arithmetic 


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Fig.  155.    Title  page  of  strigelius 


314  RARA   ARITHMETICA 

a  series  of  definitions  from  Euclid,  a  brief  treatment  of  the  operations, 
the  theory  of  proportion,  the  operations  with  fractions,  and  some  of 
the  Greek  theory  of  numbers.  One  curious  feature  of  the  book  is  the 
notation  of  Greek  fractions  after  the  Arabic  manner. 

PIERRE  SAVONNE.     Ed.  pr.  1563.  Lyons,  1571. 

A  French  arithmetician,  born  at  Avignon  c.  1525. 

Title.  '  L'Arithmetiqve//de  Pierre  Savonne,//dict  Talon, 
natif  d'Auignon//comte  de  Veniffe. // En  laquelle  font  conte- 
nues  pluiieurs  reigles  briefues  &  fubtiles,  pour  les  traffiques  de 
plufieurs  pays,  mentionnez //en  la  table  dudit  liure  :  avec  la  dif- 
ference des  poids,  aunages  //  &  monnoyes  de  chaciin  defdits 
lieiix,  alliage  de  metaux  :// neceffaire  pourtous  Maistres  de  mon- 
noyes, Orfeures  &  //  Changeurs,  avec  le  fait  &  maniement  des 
Changes  &  Ban-//ques  qui  fe  font  iournellement  a  Lyon,  8:  par 
les  places  ac-//coustumees  :  comme  Flandres,  Angleterre,  Hef- 
pagne,// Italic,  &  autres  lieux.//A  Lyon,  par  Benoist  Rigavd. 
//M.  D.  LXXL'    (F.  I,  r.) 

ColopJion.  'A  Lyon,  de  I'lmprimerie  de // Pierre  Roufsin.// 
I  571.'     (F.  148,  r.) 

Descriptioji.  8°,  10.8  X  17  cm.,  the  text  being  7.4  x  13.6  cm. 
156  ff.  (147  numb.),  28-30  11.     Lyons,  1571. 

Editions.  Paris,  1563,  4°;  ib.,  1565,  4°;  Lyons,  1571,  8° 
(here  described);  ib.,  1585;  ib.,  1588,  8°. 

This  is  one  of  the  early  French  commercial  arithmetics,  well  arranged 
but  with  no  marked  peculiarities.  It  was  so  popular  that  an  edition 
appeared  as  late  as  1672.  Like  all  arithmetics  appearing  in  Lyons  it 
devotes  much  attention  to  banking  and  exchange,  this  city  being  at  that 
time  the  commercial  center  of  France,  and  the  seat  of  one  of  the  great 
international  fairs. 

Other  works  of  1563.  Bseda,  p.  131,  1 5  2 1  ;  Boissiere,  p.  2 62,  1554; 
Feliciano,  p.  148,  1526;  Gemma,  p.  205,  1540;  Moya,  p.  310,  1562; 
Ortega,  p.  93,  1512  ;  Peletier,  p.  245,  1549;  Wolffgang  Hobel,  '  Ein 
niitzhch  Rechenblichlein  mit  viel  fchonen  Regeln  und  Fragftucken,' 
Niirnberg,  with  editions  s.  1.  (Niirnberg?),  1565  and  1577,  8°;  Simon 
Schweder,  '  Rechenbuch  von  alles  kauffmanschaft  der  Landt  auff  der 
Feder  und  Linien,'  Konigsberg,  8". 


PRINT]<:i)    HOOKS  315 

COSIMO  BARTOLI.     Ed.  pr.   1564.  Venice,  1589. 

A  Florentine  geometer,  born  in  1503;  died  in  1572.     lie  also  translated  the 
works  of  Kinaeus  (see  pp.  160,  164). 

Title.  '  Cosimo  Bartoli/Zgentirhvomo,  ct//Accademicp  Fio- 
rentino,// Del  Modo  di  Misvrare//le  diftantie,  le  fuperficie,  i 
corpi,  le//piante,  le  prouincie,  le  profpettiue,  &  //tutte  le  altre 
cofe  terrene,  che  poffo-//no  occorrere  a  gli  huomini,//  Secondo  le 
vere  regole  d'  Euclide,  &  de  gli  altri//piu  lodato  fcrittori.//  (Elab- 
orate woodcut.)  In  Venetia,  Per  Francefco  Francefchi  Sanefe. 
1589.'    (F.  I,  r.) 

Colophon.  'In  Venetia,// Per  Francefco  Francefchi  Sanefe. 
//M.  D.  LXXXIX.'     (F.  148,  r.) 

Description.  4°,  15.5  x  21.6  cm.,  the  text  being  11.2  x  15.5 
cm.     148  ff.  (6  unnumb.),  29  11.    Venice,  1589. 

Editions.  Venice,  1564,4°;  ib.,  1589,  4°  (here  described),  and 
one  edition  after  1601.  The  dedicatory  epistle  is  dated  '  il  di 
10.  di  Agofto  del  1559,'  so  there  may  have  been  an  earlier  edi- 
tion than  that  of  1564. 

Although  the  book  is  on  practical  mensuration,  the  '  libro  sesto  '  (f. 
130,  r.)  is  upon  square  and  cube  root.  The  galley  method  is  used,  and 
the  common  sixteenth-century  device  of  annexing  2  n  ciphers  in  square 
root  and  dividing  the  root  by  10"  (and  similarly  for  cube  root)  is  em- 
ployed. Bartoli  also  gives  a  table  of  squares  to  662^.  The  chapter  on 
roots  is  followed  by  one  on  the  '  Regola  delle  tre  cofe,  ouero  ([uattro 
proportionali.' 

Works  of  1564.  Albert,  p.  180,  1534;  Apianus,  p.  155,  1537; 
Benese,  p.  182,  1536;  Euclid,  p.  240,  1545;  Gutierrez  de  Gualda, 
]).  167,  1531  ;  Herbestus,  p.  303,  1561  ;  Kobel,  p.  iii,  1514;  Mariani, 
p.  181,  1535;  Medlerus,  p.  223,  1543;  Riese,  p.  139,  1522;  Sole, 
p.  146,  1526;  Tagliente,  p.  115,  15 15;  Thierfelder,  p.  391,  1587; 
Ulman,  p.  391,  1587  (Thierfelder);  Yciar,  p.  249,  1549;  Barlaamo, 
'  Arithmetica  demonstratio  eorum  quae  in  secundo  libro  f>lementorum 
(Euclidis)  sunt,'  Strasburg  (see  also  p.  343,  1572)  ;  Giovanni  Camilla, 
'  Knthosiasmo,'  Venice,  8°  (containing  a  little  work  on  arithmetic) ; 
Manuel  Fernandez  Lagasa,  '  Liljro  de  quentas,'  Salamanca,  4°,  with  a 
little  work  on  arithmetic  ;  Petrus  Nonius  (Nufiez),  '  Libro  de  Algebra 
en  Arithmetica  y  Geometria,'  with  two  editions  at  Antwerp  in  1567,  8°  ; 
the  'Opera'  of  Nonius  appeared  at  Basel  in  1592. 


3i6  RARA   ARITHMETICA 

PIERRE  FORCADEL.     Ed.  pr.  1565.  Paris,  1565. 

See  p.  2S4. 

Title.    See  Fig.  156. 

Description.  4°,  14.8  X  20  cm.,  the  text  being  9.9  x  15.9  cm. 
192  pp.  mimb.,  32-39  11.    Paris,  1565. 

Editions.  See  p.  284.  This  is  the  third  work  on  arithmetic 
published  by  Forcadel.  The  dedicatory  epistle  is  dated  '  De 
Paris,  ce  19.  de  May.  1565.'  There  was  only  one  edition,  and 
this  is  not  often  fovmd  in  dealers'  catalogues. 

In  this  book  Forcadel  does  not,  as  in  his  other  works,  take  up  the 
subject  of  counter  reckoning,  but  he  gives  a  very  satisfactory  treatment 
of  the  fundamental  operations,  the  rule  of  three,  partnership,  alligation, 
and  the  common  applications  of  the  day.  It  is  not,  however,  as  practical 
as  the  arithmetics  of  Savonne  and  Trenchant. 

ANTICH   ROCHA  de  Gerona. 

Ed.  pr.  1565.  Barcelona,  1565. 

A  Spanish  arithmetician  of  the  second  half  of  the  sixteenth  century,  born 
in  Gerona,  lecturer  at  Barcelona. 

Title.    See  Fig.  157. 

Description.  8°,  9.7  X  14.7  cm.,  the  text  being  7.5  x  12  cm. 
314  ff.  (267  numb.),  28  11.  The  dedication  by  Rocha  bears  the 
date  '  Hecha  en  Barcelona,  a.  23.  de  Nouiembre.  1564.'  Barce- 
lona, 1565. 

Editions.    There  was  no  other  edition. 

This  rare  compilation  is  based  upon  several  Italian  books,  and  the 
writer  claims  to  have  consulted  a  large  number  of  authors.  His  list 
includes  the  names  of  Feliciano,  Faber  Stapulensis,  Buteo,  Scheubel, 
Finaeus,  Ramus,  Gemma,  and  various  other  writers  of  the  time.  Rocha 
has,  however,  omitted  some  of  the  best  textbook-makers  who  preceded 
him.  The  book  is  a  fairly  complete  elementary  treatise,  the  writer  hav- 
ing taken  up  the  fundamental  operations  with  various  kinds  of  numbers, 
and  treated  each  rule  in  a  rather  scientific  way.  Although  a  consider- 
able number  of  practical  problems  relating  to  mercantile  affairs  appear 
in  the  last  half  of  the  work,  the  style  of  the  writer  is  so  prolix  that  the 
book  could  never  have  been  well  received  by  the  mercantile  classes. 
Bound  with  this  work  is  another  with  the  following  title  :   '  Compendio 


ARITHMETIOJ/E 

ENTIERE    E  T    ABREG  E  E 

DE      PIERRE      FORCADEL, 
LECTEVR    DV    ROY    ES 

MATH£MATI  QJf  E  S. 


A     PARIS. 

Chez  Charles  Pcrler,  rue  S.Iean  de  Beauuais^ 
au  Beilerophon. 

Fig.  156.   Title  page  ov  the   1565   fokcadel 


3i8  RARA   ARITHMETICA 

y  breue  //  inftruction  por  tener  Libros  de  Cuen//ta,  Deudas,  y  de  Mer- 
caduria  :  muy  prouechofo  //  para  Mercaderes,  y  toda  gente  de  negocio. 
//  traduzido  de  Frances  en   Caftellano.     En  Barcelona.//  En  cafe  de 

-^ARITHMET^ 

ca  por  Antich  Rocha 


dc  Gcrona  compucfta ,  y  de  varies 
iu£^;ores  recopilada :  pr6uech( 
para  todos  eftados  de  gentes. 


Vaanadidovn  Corapendio,para  ttncr  y  re« 

gir  los  libros  de  Cuenta-.traduzido  de  len- 

gHa  Francefa  en  Romance  Caftellano. 


EN  BARCELONA 

En  cafa  de  Claudio  Bornat,  a  la  Aguila  fuertr. 

1  S  6  i. 

Con  priuilcgio  por  dicz  anos. 

Fig.  157.   Title  page  of  rocha 

Claudio  Bornat//al  Aguila  fuerte.  1565.// Con  priuilegio  por  diez 
anos.'  (F.  286.)  This  relates,  as  the  title  suggests,  entirely  to  bookkeep- 
ing, and  is  one  of  the  earliest  treatises  upon  the  subject  in  the  Spanish 


PRINTr:!)    HOOKS  319 

language.    In  the  fourth  book  of  this  second  part  is  a  treatise  on  alge- 
bra, one  of  the  first  to  appear  in  Spain, 

Rocha  speaks  of  an  Aritmetica  by  Juan  Ventallol,  of  which  we  do 
not  know  the  date,  but  which  must  have  appeared  before  1565  (p.  298). 

ERHART  HELM.     Ed.  pr.  1565.  Frankfort,  1592. 

.V  Frankfort  arithmetician  and  ganger  of  the  middle  of  the  si.xteenth 
century. 

Title.  '1592.  Erhart  Helm/// Mathcniaticus //zu  Franck- 
furt  am  Mayn/  //  von  //  Geometrifcher  Abmeffung  der  Erdcn.// 
Item  ://  Kunftliche  Vifier  vnnd  //  Wechfelruthcn/  aul5  dem  Qua- 
drat/ //durch  die  Arithmeticam  vnnd  Geometri-//am/ gerecht 
zLimachen/ fampt  einer  luftigen  behen-//digkeit  in  Weinrech- 
nung/  //alles  durch  obgedachten // Authorem  befchrieben/ vnd 
jetzt  von  neuwem  //  widerumb  fleiffig  erfehen  vnd  cor-//rigiert.// 
Franckf.  bey  Chrift.  Egcn.  Erben.  1592.'    (F.i,  r.) 

Description.  8°,  9.5  X  15.4  cm.,  the  text  being  6.5  X  12.2  cm. 
23  ff.,  26-27  11.  Bound  with  Adam  Riese's  arithmetic  of  1592 
(p.  143).     Frankfort,  1592. 

Editions.  The  work  appears  in  the  1565  edition  of  Adam 
Riese  (p.  142),  but  not  with  a  separate  title  page.  I  have  seen 
no  separate  edition  before  this  one  of  1592. 

This  is  a  brief  treatise  on  mensuration,  and  in  particular  on  gauging, 
but  it  contains  some  explanation  of  arithmetical  processes,  including 
the  extraction  of  roots,  and  a  table  of  square  roots  to  the  e(iuivalent  of 
three  decimal  places. 

Other  7uorks  0/1565.  Agrippa,  p.  167,  1531  ;  Apianus,  p.  62,  1496 
(Jordanus)  ;  Belli,  p.  343,  1573  ;  Cusa,  p.  43,  c.  1490  ;  Delfino,  p.  275, 
1556  ;  Fischer  (Piscator),  p.  247,  1549  ;  Gemma,  p.  200,  1540  ;  Hobel, 
p.  314,  1563  ;  Jacob,  p.  295,  1560  ;  Jordanus,  p.  62,  1496  ;  Menher, 
p.  249,  1550  ;  Padovanius,  p.  389,  1587  ;  Riese,  p.  142,  1522  ;  Savonne, 
p.  314,  1563  ;  Priscian,  Rhemnius,  Fanius,  Bseda,  Metianus,  '  Liber  de 
nummis,  ponderibus,  mensuris,  numeris,  eorumque  notis,  et  de  vetere 
computandi  ratione,  ab  Elia  Vineto  emendati,'  Paris,  8°.  (Priscian's 
'  De  figuris  et  nominibus  numerorum '  had  already  appeared  in  his 
works  published  at  Venice  in  1470,  fob,  with  later  editions  in  1488, 
1492,  1495,  1496,  1519,  1525.  Such  books,  of  which  Mr.  Plimpton 
has  several,  are  not  generally  included  in  this  list.) 


320  RARA   ARITHMETICA 

HIERONYMUS  MUNYOS.     Ed.  pr.  1566.     Valencia,  1566. 

Born  at  Valencia;  died  in  15S4.  He  was  prof essor  of  mathematics  and 
Hebrew  at  Ancona,  and  later  at  Valencia.  He  also  wrote  an  astronomical 
work.    The  family  name  is  more  strictly  Muiioz. 

Title.    See  Fig.  158. 

Description.  4°,  15  X  20.4  cm.,  the  text  being  9.3  x  14.6  cm. 
4  ff .  unnumb.  +  j"]  numb.  +  i  blank  =  82  ff.,  27-3  i  11.  Valencia, 
1566. 

Editions.  There  was  no  other  edition.  The  '  epistola'  is  dated 
'  Calendis  Aprilis,  anni  M.  D.  Lxvj.' 

The  work  consists  of  three  books.  The  first  treats  of  the  funda- 
mental operations,  including  proportion  and  some  work  in  the  Greek 
theory  of  numbers.  The  second  book  treats  of  fractions,  including 
sexagesimals,  these  being  needed  by  the  astronomers  for  whom  Munyos 
was  writing.  The  third  book  relates  to  ratio  and  proportion.  Altogether 
the  work  is  too  theoretical  to  have  much  influence  upon  the  develop- 
ment of  arithmetic. 

IAN  TRENCHANT.     Ed.  pr.  1566.  Lyons,  1578. 

A  Lyons  arithmetician,  born  c.  1525. 

Title.  '  L'Arithme-//tiqve  de  Ian  //  Trenchant,// Departie  en 
trois  //  liures.//Enfemble  vn  petit  difcours  des  Changes.//  Avec 
//  L'art  de  calculer  aux  Getons.//  Reueiie  &  augmentee  pour  la 
quatrieme  edition, //de  plufieurs  regies  &  articles, //par  lAutheur. 
//  A  Lyon,//  par  Michel  love,//  et  lean  Pillehotte.//  a  I'enfeigne 
du  lefus.//  1578.// Auec  priuilege  du  Roy.'    (P.  i.) 

Description.  8°,  10. 4  X  15.5  cm.,  the  text  being  7  X  13.7  cm. 
375  pp.  numb,  -f  5  unnumb.  =  380  pp.,  30-31  11.    Lyons,  1578. 

Editions.  Lyons,  1566;  ib.,1571;  ib.,  i  578, 8° (here described). 
The  dedication  of  the  1578  edition  bears  the  date  '  De  Lyon  ce 
9.  de  luillet  1571,'  and  this  is  often  given  as  the  date  of  the  first 
edition.  Although  this  is  described  on  the  title  page  as  the  fourth 
edition,  I  know  of  only  two  earlier.  Cantor  says  that  not  less 
than  six  editions  were  published  at  Lyons  from  1588  to  1602, 
and  I  have  seen  mentioned  editions  of  1608, 1610, 1632,  and  1643. 

Trenchant  was  one  of  the  best  of  the  sixteenth-century  textbook- 
makers  of  commercial  arithmetic  in  France.    His  work  is  divided  into 


PRINTI-:!)    HOCJKS 


321 


INSTITVTIONES 

ARITHMETICAE    AD    ^ER. 

CiPIENDAM      A  f  T  R  O  I,  O  G  I    A  M     BT 

Matheinaticas  facultatcs  neccffarias. 

UVCTOJtE 

Htercnymo  Munyos  Valentino  Hehdica  tij9r 

^u<e ^aricer  at^  Mathematumin^y'* 

mnafio  XJalentinofuhlico 

frofejfor^^ 


cV*^ 


V  A  L  E  N  T  I  AE. 

Ex  tjrpographia  loannis  Mey. 
Anao  1566. 


ZJJ 


Fk;.  158.   Title  page  of  munyos 


322  KARA   ARITHMETICA 

three  books,  the  first  dealing  with  the  fundamental  operations  with 
integers  and  fractions,  and  containing  a  considerable  number  of  applied 
problems.  The  second  book  treats  of  the  rule  of  three  in  its  various 
forms  together  with  such  applications  as  barter,  partnership,  commis- 
sion, and  alloys.  The  third  book  treats  of  the  properties  of  numbers, 
including  figurate  numbers,  roots,  and  progressions,  and  has  some  work 
on  discount,  together  with  a  few  recreations.  In  the  1578  edition  the 
third  book  is  followed  by  a  chapter  on  exchange,  and  an  explanation  of 
the  method  of  calculating  with  counters. 

GEORGIO  LAPAZZAIA.     Ed.  pr.  1566.  Naples,  1569. 

Lapezaja,  Lapizaya,  Lapazaia.  Probably  a  resident  of  Naples,  but  born 
at  Monopoli,  a  town  in  the  province  of  Bari,  on  tlie  Adriatic.  He  was  a 
priest,  and  wrote  only  the  worli  here  described. 

Title.    See  Fig.  159. 

Colophon.  '  In  Napoli // Apreffo  Mattio  Cancer.  M.D.LXIX. 
//Con  Priuilegio  per  anni  diece.//Marius  Carrafa  Archiepifcopus 
Neapolitanus.'     (P.  262.) 

Description.  4°,  14.  i  X  21  cm.,  the  text  being  10.8  X  16.9  cm. 
13  pp.  unnumb.  +  250  numb.  =  263  pp.,  29  11.    Naples,  1569. 

Editions.  Naples,  1566,  Latin  edition  with  the  title  'De  fami- 
liarite  arithmetical  et  geometriae,'  this  date  appearing  also  in  the 
privilege,  'Datum  Neapoli  die  vltimo  lulii  M.D.LXVI;'  ib.,  1566, 
Italian  edition;  ib.,  1569,  4°  (here  described);  1575;  Naples, 
1590,  4°  (p.  324).  There  were  several  editions  after  1600,  one 
appearing  as  late  as  1784.     For  the  1601  edition  see  p.  324. 

The  book  has  nothing  to  commend  it  except  its  popular  style. 
Lapazzaia  begins  with  the  fundamental  operations  with  integers,  and 
then  treats  of  ratio,  fractions,  and  progressions.  He  considers  also  the 
rule  of  three,  the  rule  of  five,  interest,  exchange,  partnership,  alligation, 
rule  of  false,  and  the  extraction  of  roots.  The  last  part  of  the  work  is 
on  mensuration. 

Other  works  0/1566.  BeUi,  p.  343,  1573  ;  Fischer  (Piscator),  p.  247, 
1549  ;  Gemma,  p.  200,  1540  ;  Herbestus,  p.  303,  1561  ;  Monzd,  p.  292, 
1559  ;  Georg  Meyer,  '  Rechenbiichlein  defs  Silberkauffs  und  gemachter 
Arbeit,'  Augsburg,  16°;  Mathaus  Nessen  (Nesse),  '  Zwei  neue  Rechen- 
biicher,'  Breslau,  8°;  Johannes  de  Segura,  '  Mathematicae  quaedam 
selectae  propositiones,'  Alcala,  4°,  and  *  Compendium  Arithmeticae  et 
Geographiae  partis,'  Alcala,  4°. 


PRINTED    HOOKS 


323 


DARITMETTCA  E  GEO- 

METRIA  DELUABBATE   GEOR* 
GlO  iAPAZZAIA    WONOPOLITANO. 


Fig.  159.   Title  page  of  the   1569  lapazzaia 


324  RARA   ARITHMETICA 

GEORGIO   LAPAZZAIA.     Ed.  pr.  1566.  Naples,  1590. 

See  p.  322. 

Title.  '  Opera  //  terza  //  de  Aritmetica  //  et  Geometria.  // 
Dell'abbate  Georgio  Lapazaia//da  Monopoli.//Intitolata  il  Ra- 
maglietto.//In  Napoli,//Apreffo  gli  Eredi  di  Mattio  Cancer.// 
M.  D.  LXXXX.'    (P.  I.) 

Colophon.  'In  Napoli // Apreffo  gl' Eredi  di  Mattio  Cancer.// 
M.  D.  LXXXX.'    (P.  176.) 

Description.  4°,  13.9  X  18.7  cm.,  the  text  being  10.8  X  16.5 
cm.  I  p.  unnumb.+ I  blank +174  numb.  =  176  pp., 29 11.  Naples, 
1590. 

Editions.    See  p.  322. 

Although  bearing  a  different  title,  this  is  merely  a  revision  of  the 
1566  work,  with  a  slight  variation  in  the  problems.  The  preface  is  dated 
1569,  when  the  second  edition  of  the  Italian  version  appeared,  but  it 
is  not  found  in  that  edition.  The  tide  '  Opera  terza'  means  simply  the 
second  revision,  or  the  third  writing  of  the  book.  Lapazzaia's  work  shows 
the  increasing  attention  given  by  the  Church  schools  to  the  business 
needs  of  the  people. 

GEORGIO   LAPAZZAIA.     Ed.  pr.  1566.  Naples,  1601. 

See  p.  322. 

Title.  '  Libro  //  d' Aritmetica  //  e  Geometria,  //  dell' Abbate 
Giorgio  Lapazzaia  //  Canonico  Monopolitano,  e  Protonotario 
Apoftolico.// Nouamente  in  queft'vltima  imprefsione  efpurgato 
da  molti  errori,  &  arric-//cbite  d'vna  Prattica  d'Abbaco,  non 
meno  vtilifsima,  che  neceffaria.// Al  Signor  Diego  d'Aldana,// 
Prefidente  della  Regia  Camera  della  Summ.  per  Sua  Maiefta.// 
(Large  woodcut,  coat  of  arms.)  In  Napoli, //Apreffo  Tarquinio 
Longo.    MDCL'    (P.  i.) 

Colophon.  'In  Napoli,// Apreffo  Tarquinio  Longo.  MDCL' 
(R  215.) 

Description.  4°,  14.9  X  20  cm.,  the  text  being  10.7  X  15.8  cm. 
216  pp.  (3  blank,  8  unnumb.),  30-38  11.     Naples,  160 1. 

See  p.  322, 


PRINIICI)    HOOKS  325 

NICOLAUS  PETRI.     Ed.  pr.  1567.  Amsterdam,  1635. 

Born  at  Deveiiter.    He  taught  at  Amsterdam  from  1567  to  15S8.    lie  also 
wrote  on  algebra  and  astronomy. 

Title.  'Practicque //Om  te  leeren//Rceckenen/ Cyphercn // 
ende  Boeckhouwen/  met  die  regel  Cof.s/  //ende  Gcomctrie/  fecr 
profijtelijcken  voor  alien  //  koop-luyden.  Van  nieus  gecorrigeert 
//ende  vermeerdert/ //Deur  Nicolaum  Petri  Daventrienfem.// 
L'homme  propofe,  Et  dieu  dispofe.  A"  1603.//  (Woodcut  of 
author.)  t'Amstelredam,//Voor  Ilendrick  Laurentfz.  Boeckver- 
cooper  op  het// water  int  Schrijf-boeck,  Anno  1635.'    (F.  i,  r.) 

Description.  8°,  10.3  X  16.3  cm.,  the  text  being  8.9  x  14.4  cm. 
6  ff.  unnumb.  +  281  numb.  =  287  ff.,  23-32  11.  Bound  with  this 
is  '  I  596.//  lournael-Boeck  //  gheteeckent  met  die  //  Letter  // 
Anno    M.  DC.  XXXV.,'  10  ff.    Amsterdam,  1635. 

Editions.  From  the  preface  it  appears  that  the  work  was  first 
published  in  1567.  The  other  sixteenth-century  editions  were 
Amsterdam,  1576,  8°;    1583;    1591;  and  Alkmaar,  1596,  8°. 

The  arithmetic  is  of  the  ordinary  Dutch  type.  It  uses  only  the 
galley  form  of  division,  and  its  chief  value  to  the  student  of  history  lies 
in  its  business  problems. 

Other  works  of  ijSy.  Agrippa,  p.  167,  1531  ;  Borghi,  p.  16,  1484  ; 
Cataneo,  p.  244,  1546  ;  Gemma,  p.  205,  1540  ;  Mariani,  p.  181,  1567  ; 
Nonius  (Nuiiez),  p.  315,  1564;  Ramus,  p.  263,  1555;  Tagliente, 
p.  115,  15 15  ;  Conradus  Dasypodius,  '  Logistica,'  Strasburg,  8°;  Guill. 
de  la  Toissoniere,  '  Compost  arithmetical,'  Lyons. 

STEFANO  GHEBELINO.     Ed.  pr.  1568.  Brescia,  1568. 

An  arithmetician  of  Brescia,  of  the  second  half  of  the  si.xteenth  century. 

Title.    See  Fig.  160. 

Colophon.  *  In  Brescia,// Apresso  Vincenzo // di  Sabbio.// 
M.  D.  LXVIII.'     (F.  34,  r.) 

Description.  4°,  13.5  x  19.5  cm.,  the  text  being  10.  i  x  16.5 
cm.     II  ff.  unnumb.  +  33  numb.  =  44  ff.,  37  11.    Brescia,  1568- 

Editions.    There  was  no  other  edition. 

The  dedicatory  epistle  shows  that  this  work  was  written  at  Brescia 
in  October,  1568.    It  is  composed  almost  entirely  of  tal)les  for  the  use 


326 


RARA   ARITHMETICA 


Fig.  i6o.    Title  page  of  ghebelino 


I 


PRINTED    HOOKS  327 

of  merchants  and  bankers  in  the  north  of  Italy.  There  is  some  explan- 
atory matter  in  the  beginning  relating  to  the  arithmetic  of  exchange, 
but  the  work  can  hardly  be  called  a  school  textbook.  Riccanli  speaks 
of  it  as  '  uno  dei  primi  esempi  di  tavole  di  conti  fatti.' 

HUMPHREY  BAKER.     Ed.  pr.  1568.  London,  1580. 

Bom  at  Loiulon  ;  died  after  15S7. 

Title.  'The  Well  fpring  of // Sciences.// Which  teacheth  the 
perfect // worke  and  practife  of  Arith-//nieticke,  both  in  whole 
Ntim-//bcrs  and  Fractions :  fet  //  forthe  by  //  Humf rey  Baker  // 
Londoner,  i  562.// And  nowe  once  agayne  pertifed  //augmented 
and  amended  in  all  //  the  three  partes,  by  the  sayde  //Aucthour: 
where  tmto  he  //  hath  alfo  added  certein  //  tables  of  the  agree-// 
ment  of  mcafures //and  waightes //of  diuers  places  in  Europe, 
//  the  one  with  the  other,  as  //  by  the  table  following  //  it  may 
appeare.// 1580.'     (F.  i,  r.) 

Description.  8°,  9.9  X  14  cm.,  the  text  being  5.6  X  11. 2  cm. 
227  ff.  (28  unnumb.,  last  folio  missing),  24-26  11.    London,  1580. 

Editions.  London,  1568,  8°  (written  in  1562,  which  explains 
the  date  on  the  title  page);  ib.,  1574,  8°;  ib.,  1580,  8°  (here 
described);  ib.,  1583,  8°;  ib.,  1591,  8°.  There  were  several 
editions  after  1601  (see  pp.  328,  329). 

For  a  long  time  Baker's  arithmetic  was  the  only  English  rival  to 
Recorde's  '  Ground  of  Artes '  (see  p.  213),  and  it  was  in  many  respects 
better  than  that  popular  work.  This  edition  is  more  complete  than  that 
of  1568,  the  book  having,  as  the  author  states,  been  rewritten.  In 
*  The  Prologue  to  the  gentle  Rerder  '  he  says  :  *  Hauing  fometime  now 
twelue  yeres  fithence  (gentle  Reader)  publifhed  in  print  one  Englifhe 
boke  of  Arithmetick,  conteyning  as  I  suppose,  fundry  necfsarie  and 
profitable  documentes  for  fuch  as  are  A"villing  to  attayne  any  knowlege 
therein.  I  have  bene  often  fmce  that  time,  and  of  very  late  alfo, 
requefted  by  fundry  of  my  friendes  to  perufe  the  fame  worke,  and  as  I 
fhold  nowe  iudge  it  expedient,  to  adde  fomething  more  therevnto,  and 
to  amplifye  the  fame.'  He  complains  of  the  criticism  of  foreigners  that 
English  arithmetic  is  not  as  advanced  as  that  on  the  continent :  '  For 
when  I  perceyued  the  importunitie  of  certayne  ftraungers  not  borne 
within  this  lande,  at  this  prefent,  and  of  late  dayes  fo  farre  proceeding, 
that  they  aduaunced  and  extolied  them  felues  in  open  talke  and  writinges, 


328  KARA   ARITHMETICA 

that  they  had  attayned  fuch  knowledge  and  perfection  in  Arithmetike, 
as  no  englifh  man  the  Uke  :  Truly  me  thought  that  the  fame  reporte 
not  only  tended  to  the  (disprayfe)  difpraife  our  Countreymen  in  gen- 
eral :  But  touched  efpetially  fome  others  &  me,  that  had  trauailed 
&  written  publiquely  in  the  fame  facultie.  For  vnto  this  fame  effecte 
they  haue  of  late  paynted  the  corners  and  poftes  in  euery  place  within 
this  citie  with  their  peeuifhe  billes,  making  promife  and  bearinge  men 
in  hande  that  they  coulde  teache  the  fumme  of  that  Science  in  breefe 
Methode  and  compendious  rules  fuch  as  before  their  arriuall  hath  not 
bene  taughte  within  this  Realme.'  These  words,  and  others  in  the  same 
strain,  give  an  interesting  picture  of  English  arithmetic  in  1580,  and  of 
the  work  of  the  teacher  at  that  time.  The  criticism  was  a  just  one, for 
the  Dutch,  French,  Germans,  Spanish,  and  Italians  were  much  ahead 
of  the  English  at  that  period  in  the  matter  of  arithmetic. 

Baker  follows  the  continental  models,  giving  the  usual  operations 
and  the  applications  to  '  Marchandife,'  '  Felowfhip,'  barter,  alligation, 
false  position,  and  the  like.  He  closes  his  text  with  '  Quftions  of 
Paftime.'  He  still  uses  duplation,  generally  uses  the  form  '  substraction  ' 
(following  the  Dutch  books  of  the  time),  and  makes  relatively  little  of 
'  Deuifion,'  which  he  treats  by  the  usual  galley  method,  but  he  succeeds 
in  producing  a  fairly  practical  mercantile  book. 

Other  works  0/1568.  Anianus,  p.  32,  1488  ;  Delfino,  p.  275,  1556  ; 
Gemma,  p.  200,  1540;  Gulfferich,  p.  257,  1552;  Lonicerus,  p.  253, 
1551  ;  Lossius,  p.  290,  1557;  Riese,  p.  139,  1522;  Sfortunati,  p.  174, 
1534;  Mauritius  Steinmetz,  'Aritmeticae  praecepta  in  quaeftiones 
redacta,'  Leipzig,  sm.  8°. 

HUMPHREY  BAKER.     Ed.  pr.  1568.  London,  1659. 

See  p.  327. 

Title.  '  The  //  Wel-spring  //  of  //  Sciences  :  //  teaching  The 
perfect  Work  and  Practice  of  //  Arithmetick,//  both  in  Numbers 
and  Fractions.//  Set  forth  by//  Humphrey  Baker  //  Londoner.// 
And  now  again  Perufed,  Augmented,  and //Amended  in  all  three 
Parts,  by //the  faid  Authour.// Whereunto  are  added  certain 
Tables  of //the  agreement  of  Meafures  &  Weights //of  divers 
places  in  Europe,// the  one  with  the  other,  as  by  the  Table  appear- 
eth.//  London,//  Printed  for  A.  Kemb,  at  St.  Margarets  Hill  in 
//  Southwark,  to  bee  fold  by  Tho.  Brewfter,// at  the  three  Bibles 
in  Pauls  Church-yard  :  1659.'     (P.  i.) 


PRINTED    HOOKS  329 

Description.   8°,  8.4  X  13.6  cm.,  ihc  text  being  6.3  X  i  1.2  cm. 
366  pp.  (54  unnumb.),  26-27  "•     I^ondon,   1659. 
Editions.    See  p.  327. 

This  is  not  materially  different  from  the  1580  edition  (p.  327).  The 
publisher  says  that  '  The  friendly  Reader  may  pleafe  to  take  notice  that 
in  this  Impreffion  of  1659,  the  whole  Book  hath  been  revifed,  every 
Queftion  therein  examined,  the  Faults  that  were  committed  in  former 
ImpreiTions,  Corrected,  the  whole  reftored  to  its  first  integrity.'  In 
spite  of  this  statement,  there  is  little  improvement  in  the  book  either  in 
methods  of  operating  or  in  symbolism.  It  is  interesting  to  read  that 
'  The  Fractions  [in  the  tables  at  the  end  of  the  book]  which  before 
were  in  the  common  way  (and  fo  the  figure  being  fmall  in  many  not 
difcerned)  are  put  into  the  decimal  parts,  and  fo  the  fame  with  the 
integral,  but  farre  more  true  than  the  Common  Fractions  can  exprefs 
it  in  one  figure,  and  if  in  the  common  it  be  expreft  in  many,  (as  it  must 
be,  if  true)  then  the  decimal  is  far  more  eafie,  becaufe  the  Denominator 
is  one  and  the  fame  to  all,  whereas  the  other  is  differing.'  In  fact, 
very  slight  knowledge  of  decimals  is  shown,  and  when  they  are  employed 
the  bar  is  generally  used  instead  of  the  point. 

HUMPHREY  BAKER.     Ed.  pr.  1568.  London,  1687. 

See  p.  327. 

Title.  'Licenfed,//Feb.  28,  i68''.//Rob.  Midgley.'  (P.  4, 
first  page  of  print.)  '  Baker's  //  Arithmetick  ://  Teaching  //  The 
perfect  Work  and  Practice  of //Arithmetick  both  in //Whole 
Numbers  &  Fractions.//  Whereunto  are  Added  //  Many  Rules 
and  Tables  of // Intereft,  Rebate,  and  Purchafcs,  &  c.// Also// 
The  Art  of  Decimal  Fractions,// intermixed  with  Common  Frac- 
tions, for  the  //  better  Underftanding  thereof.//  Newly  Corrected 
and  Contracted,  and //made  more  plain  and  eafie //By  Henry 
PhilHppes.// London.// Printed  by  J.  Richardfon  for  William 
Thackery  at  the //Angel  in  Duck-Lane,  and  Matthew  Wotton 
at  the  Three //Daggers  in  Fleet  ftreet,  and  George  Conyers  at 
the // Ring  without  Ludgate,  1687.'     (P.  5.) 

Description.  \2° ,  8.3  x  14.3  cm.,  the  text  being 6.8  X  12.8  cm. 
8  pp.  blank  +  lO  unnumb.  +  228  numb.  =  246  pp.,  32-39  11.  Lon- 
don, 1687. 


330  RARA   ARITHMETICA 

Editions.    See  p.  327. 

Phillippes,  the  editor,  pays  a  deserved  tribute  to  Baker  in  his  letter 
'  To  the  Reader.'  He  begins  as  follows  :  '  This  little  Book,  as  it  was 
one  of  the  firft,  fo  it  is  one  of  the  beft  of  this  Subject,  and  hath  had  as 
good  Acceptance,  as  any  other ;  which  may  appear  by  the  often  Impref- 
fions  of  it.  Indeed  as  long  as  the  Author  lived,  he  was  careful  to  be 
ftill  adding  and  correcting  it :  and  though  he  be  dead,  yet  his  Book  is 
thought  worthy  to  live,  and  not  only  to  live,  but  to  flourifh.'  It  certainly 
speaks  well  of  the  book  that  this  edition  should  have  been  published 
121  years  after  the  first  one  appeared.  The  treatment  of  decimal  frac- 
tions is  very  satisfactory,  and,  of  course,  is  not  found  in  the  original 
edition.  These  fractions,  the  necessity  for  which  became  apparent  in 
the  sixteenth  century,  were  first  scientifically  treated  at  any  length 
in  a  work  by  Stevin,  published  in  1585.    (See  p.  386.) 

PETRUS  RAMUS.     Ed.  pr.  1569.  Basel,  1569. 

See  p.  263. 

Title.  '  p.  Rami  Arith-//meticae  libri  //  dvo  :  Geometriae  // 
septem  et  viginti.// (Woodcut.)  Basiliae,  per  Evsebivm // Epif- 
copium,  &  Nicolai  fratris  haeredes.//  Anno  M.  D.  LXIX.'    (P.  i .) 

Dcscriptio7i.  4°,  17.5  X  23.3  cm.,  the  text  being  12.3  x  18.2 
cm.     198  pp.  (2  blank,  6  unnumb.),  42  11.    Basel,  1569. 

Editions.  Basel,  1569,4°  (here  described);  Paris,  1577,  8° 
(p.  331);  Basel,  1580,  4°  (p.  331);  Paris  (Stadius  edition),  1581, 
12°;  Frankfort  (Schonerus  edition),  1586,  8°  (p.  331)  ;  ib.,  1591, 
8°  (Stegerus  edition) ;  ib.,  1 592,  8°  (Schonerus) ;  ib.,  i  596,  8°  (the 
Snellius  and  Schonerus  '  Explicationes,'  p.  333);  ib.,  1599,  fol. 
(Schonerus,  p.  333) ;  Lemgo,  1 599,  4°.  There  was  also  published 
at  Paris  in  1562,  and  in  two  editions  the  same  year,  an  '  Arith- 
metica,'  in  two  books,  without  the  author's  name,  attributed  to 
Ramus,  but  I  do  not  know  whether  it  is  the  same  as  this  work. 
There  was  also  an  English  edition  of  '  The  Art  of  Arithmeticke 
in  whole  numbers  and  fractions  ...  by  P.  Ramus  .  .  .  translated 
...  by  William  Kempe,'  London,  1592,  8°. 

This  is  a  better  book  than  the  '  Libri  Tres'  of  1555.  Ahhough  it  is 
too  theoretical  to  have  met  the  commercial  needs,  it  is  a  nearer  approach 
to  a  practical  work  than  its  predecessor. 


PRINTED    BOOKS  331 

PETRUS   RAMUS.     Kd.  pr.   1569.  Paris,  1577. 

See  p.  263. 

Title.  'Petri  Rami // Profcssoris  Rej^ni,// Arithmcticcc // li- 
bri  dvo.// Parisiis,// Apud  Dionyfuim  Vallcnfcm,  fub // Pegafo, 
in  vico  Bellouaco.//  1577.'    (P.  i,  r.) 

Description.  8°,  10.3  X  16  cm.,  the  text  bein<^  7.1  x  13.1  cm. 
97  ff.  (2  blank,   1  imnumb.)  with  chart;    32  11.     Paris,  1577. 

See  p.  330. 

PETRUS  RAMUS.     Ed.  pr.  1569.  Basel,  1580. 

See  p.  263. 

Title.  '  P.  Rami  //  Arithmeticae  //  libro  dvo  //  Geometriae  // 
septem  et  viginti.//  (Woodcut.)  Basileae,  per  Evsebivm  //  Epifco- 
pium,  &  Nicolai  fratris  ha;redes.//  M  D  LXXX.'    (P.  i.) 

Colophon.  '  Basileae,  per  Evsebivm  Episco-//pium,  &  Nicolai 
fratris  h^redes.  Anno // M.  D.  LXXX.'    (P.  192.) 

Description.  4°,  15.9  X  21.2  cm.,  the  text  being  i  i.i  X  17  cm. 
200  pp.  (9  unnumb.),  28-35  ^l-    Basel,  1580. 

See  p.  330. 

PETRUS  RAMUS  and  LAZARUS  SCHONERUS. 

Ed.  pr.  1569.  Frankfort,  1586. 

See  p.  263. 

Title.    See  Fig.  161. 

Description.  8°,  10.5  x  17.3  cm.,  the  text  being  7  x  13.1  cm. 
16  pp.  unnumb.  +  406  numb.  =  422  pp.,  26-32  11.  Frankfort, 
1586. 

Bound  with  this  is  '  P.  Rami,  Regii  //  Eloquentioe  et  //  Philo- 
sophiae  Pro-//fefforis,  liber  de  moribus//veterum  Gallorum,//ad 
//  Carolum  Lotharingum  //  Cardinalem.//  Parisiis,//  Apud  An- 
dream  Wechelum.//  1562.// Cum  privilcgio  Regis.' 

Editions.  See  p.  330.  The  commentary  of  Schonerus  also 
appeared  with  that  of  Snellius  in  1596  (p.  333),  and  without  the 
latter  in  1599  (p.  333).  There  were  also  editions  by  Steger  pub- 
lished at  Leipzig  in  1591  and  at  Frankfort  in  1592. 


332 


KARA   ARITHMETICA 


PETRI  RAMI 

ARITHMETICES    LI- 

BRI    DVO,    ET     AlGEBR^E 

totidemiaLAZARo  Schonero 

emendati  &explicati. 

Eiufdem  S  c  H  o  n  e  r  i  If  hi  duo :  alter ,  De 
J^umerisfguratis;  altera  De  Logiflica 
fixagemrk* 


Francofurdi 
Apud  heredes  Andreas  Wecheli, 

MDLXXXVr. 
(iim  S.CafcJl^aiefiatupriuflegio  adfixenninm . 

Fig.   i6i.    Title  page  of  the   1586  ramus 


PRIN  TKI)    BOOKS  333 

The  first  of  these  works  is  one  of  several  commentaries  on  tlir  theo- 
retical arithmetic  of  Ramus  (p.  263).  It  is  more  practical  than  that  of 
Snellius  (mentioned  below),  giving  the  various  operations  and  making 
an  attempt  at  introducing  some  commercial  problems. 

In  the  first  part  are  included  two  works  by  Schonerus,  '  I)e  nume- 

ris  figuratis  Lazari  Schoneri  liber '  and  '  Lazari  Schoneri  De  logistica 

sexagenaria  liber.'    The  former  is,  as  the  title  suggests,  a  treatise  on 

the  Greek  theory  of  numbers,  and  the  second  is  on  the  sexagesimal 

fractions  used  by  the  astronomers.    Schonerus  writes  his  sexagesimals 

Ilae    lee      o        I        II      ,  ,   .,  ,         .      ^  ,  -<^  ,   4o 

thus  :  for     3-60-  +  39-60  +  4"  +  —  +  7— ,  • 

3.      39.    40.     20.     40.,  00       00 

This  is  one  of  the  early  approaches  to  our  symbols  °,  ',  ". 

PETRUS  RAMUS  and  RUDOLPHUS  SNELLIUS. 

Ed.  pr.  I  569.  Frankfort,  i  596. 

See   p.   263.    Snkllius,   l)orn    at    Oudewater,   October  8,    154^';   died  at 
Leyden,  March  2,  161 3. 

Title.    See  Fig.  162. 

Description.  8°,  9.6  X  16  cm.,  the  text  being  6.6  x  12.8  cm. 
3  pp.  unnumb. +  154  numb.  =  157  pp.,  29-30  11.  PVankfort, 
1596.  Boimd  with  this  are  '  Rvdolphi // Snellii  in //P.  Rami 
Geome-//triam  PraeIe-//ctiones,'  and  '  Rudolphi  //  Snellii  in 
Sphaeram  Cor-//nelii  Valerii // praelectiones,'  both  of  1596. 

Editions.   See  p.  330.  The  first  edition  of  Snell's  commentary. 

Like  the  arithmetic  of  Ramus,  this  work  is  theoretical  rather  than 
practical.  Only  the  prominence  of  Ramus  could  have  justified  such 
efforts  as  these  of  Snellius,  Salignacus,  and  Urstisius. 

PETRUS  RAMUS.     Ed.  pr.  1569.  Frankfort,  1599. 

See  p.  263. 

Title.  '  Petri  Rami  //  Arithmeticae  //  libri  dvo :  Geomctriae  // 
septem  et  viginti.//  A  Lazaro  Schonero  recogniti  &  aucti.//  Fran- 
cofvrti,//  Apud  Andreae  Wecheli  heredes,//  Claudium  Marniimi, 
&  loannem  Aubrium.//M.  D.  XCLX.'     (P.  3.) 

Description.  Fob,  17  x  22.5  cm.,  the  text  being  11.9X  17.6 
cm.  244  i^p.  in  the  arithmetic,  184  \)\>.  in  the  geometry,  39  11. 
Frankfort,  1599.     Bound  with  this  is  the  geometry  of  Ramus. 


334 


RARA   ARITHMETICA 


IN  P.  RAMI  ARITH- 
METIC A  M 

RVDOLPHI 

SN  E  L  L I I 

Explicationes  le(aiflimae: 

LAZARI  SCHOT(ERIy  BERT^Hi 

Saligmci ,  ^  Chnfliam  Vrfiifii ,  com- 

mentAttonthmfapm  lo- 

cupleeatdi. 


FRANCOFVRTI 

Ex  Officina  Typogtaphica  loannis  Sautii, 
impenfis  haeredum  Petri  Fifcheri. 

M.    D.    XCVl. 

Fig.  162.   Title  page  of  the   1596  ramus  and  snellius 


PRINTED   BOOKS  335 

PETRUS   RAMUS.    Ed.  pr.  1569.  Basel,  1569. 

See  p.  263. 

Title.    See  Fig.  163. 

Coloplion.  'BasilecX,  per  Evsebivm  Episco-//pium,  &  Nicolai 
fratris  hxredes.  Anno  Salutis  humanrc//M.  D.  LXIX.'  (P.  190 
of  the  geometry,  bound  with  the  above,  or  534  of  the  entire  book.) 

Description.  4°,  17.5  X  23.3  cm.,  the  te.xt  being  12.3  X  18.2 
cm.  16  pp.  unnumb.  +  320  numb.  =  336  pp.  of  the  above  (not 
including  the  rest  of  the  work),  42  11.    Basel,  1569. 

Editions.  Basel,  1569,  4°  (here  described);  ib.,  1578  (Schone- 
rus  edition);  Frankfort  (also  Schoncrus  edition),  1599,  4°.  The 
first  three  books  also  appeared  at  Paris  in  i  567,  8°,  under  the 
title  'Prsemium  Mathematicarum.' 

An  extensive  and  tiresome  treatise  on  the  philosophy  of  elementary 
mathematics  in  general. 

THOMAS  DE  MERCADO. 

Ed.  pr.  1569.  Salamanca,  1569. 

A  Spanish  priest  of  the  middle  of  the  si.xteenth  century. 

Title.    See  Fig.  164. 

Description.  8°,  13.5  X  19.4  cm.,  the  text  being  11.5  X  16  cm. 
277  ff.  (29  unnumb.),  32  11.    Salamanca,  1569. 

Editions.    There  was  no  other  edition. 

This  is  not  a  textbook  on  arithmetic,  but  a  treatise  on  the  applica- 
tions of  the  subject  to  mercantile  affairs.  It  is  so  prolix  and  theoret- 
ical that  it  was  never  republished.  That  it  is  the  first  edition  appears 
from  '  La  Tassa '  and  from  the  dedication  to  the  king,  the  former  being 
dated  October  6,  1569,  and  the  latter  May  6  of  the  same  year.  The 
license  is,  however,  dated  August  13,  1568,  and  one  of  the  decrees 
May  9,  1568.  Although  the  work  professes  to  be  of  a  mercantile  char- 
acter, it  is  too  ponderous  in  style  for  the  purpose  for  which  it  was 
intended.  It  is  interesting  historically  because  in  several  chapters  the 
author  has  considered  the  development  of  arithmetic  and  of  mercantile 
customs.  It  is  also  interesting  because  of  its  reference  to  the  recently 
awakened  commerce.  For  example,  chapter  13  has  the  title  '  De  los 
tratos  de  Indias,  y  tratantes  en  ellos.'  Chapter  16  is  also  suggestive  of 
the  methods  of  trade  of  the  period,  the  title  being  '  De  los  baratas  y 


336  RARA   ARITHMETICA 

hyryyr    A'^rttf^     Itmrpt^n   ^»^/»-*^tM4     Aiii^^€S9v 

p.   RAMI   SCHO'^-i:? 

LARVM     MATHEMATICAL 

TRIGINTA. 


BASILS  AE,      PER      EVSEBIVM      EPISCOPIVM, 
O"  Njcoldi  Trutrif  htrtdts. 
ANNO     M.    D.    LXIX  ry*,^ 

Fig.  163.    Title  page  ok    the  Li/>n  viii^s  rf  tih^nita  ok  kamus 


TRATOS  Y  CON- 

TRATOS  DE  MERCADERES 

y  tratantes  difddidosy  determinados,por 

el  Padre  Prefentado  Fray  Thomas 

de  Mercado,de  la  orden  de  los 

Predicadores. 


Conlicenciay  priuilegio  real. 


EN  SAL  A  M  ANCJ. 

For  aMathias  (jaFl.  aAno  de 

Efta  caflado  en    cincorealcs. 

F'iG.    164.     TiTLK    PACK    OF    MKKCADo 


338  KARA   ARITHMETICA 

de  la  nauegacion  de  las  Indias.'  The  treatment  of  interest  and  exchange 
is  more  extensive  than  usual,  the  former  having  been  by  no  means 
popular  in  Spain.  The  last  part  of  the  book  is  devoted  entirely  to  legal 
questions. 

Other  zvorks  of  i^dg.  Belli,  p.  343,  1573;  Camerarius,  p.  263, 
1554  ;  Gemma,  p.  200,  1540  ;  Herbestus,  p.  303,  1561  ;  Jacob,  p.  298, 
1560;  Lapazzaia,  p.  322,  1566  ;  Lossius,  p.  290,  1557  ;  Mariani,  p.  181, 
1535;  Monz6,  p.  292,  1559;  Urstisius,  p.  361,  1579;  Jacob  Frey, 
'  Exempelbiichlein  allerley  Kaufmannshiindel,'  Niirnberg  (there  was  also 
an  Augsburg  edition  of  1 603, 1 6°) ;  Adriaen  van  der  Gucht,  '  Cyferbouck,' 
Bruges,  4°  ;  James  Peele,  '  The  pathewaye  to  perfectnes  in  th'  accomptes 
of  debitour  and  creditour,'  London,  fol.  (second  edition). 

HIERONYMUS  CARDANUS. 

Ed.  pr.  1570.  Basel,  1570. 

See  p.  193. 

Title.    See  Fig.  165. 

Colophon.  '  Basileae,//  ex  officina  Henricpetrina,  Anno  //  Salv- 
tis  M.  D.  LXX.  Mense  //  Martio.'     (P.  1 1 1  of  the  third  part.) 

Description.  Fol.,  20.4  X  30  cm.,  the  text  being  13.2  X  24  cm. 
291  pp.  (4  blank,  16  unnumb.)  in  this  book.  Bound  with  this  is 
the  *Ars  Magna'  (second  edition),  163  pp.  numb.,  and  the  '  De 
aliza  regvla  liber,  hoc  est,  algebraicae  logifticse  fuae  .  .  .,'  120  pp. 
(Ill  numb.),  41  11.    Basel,  1570. 

Editions.  There  was  no  other  separate  edition  of  the  '  Opus 
Novum.' 

This  work  is  particularly  interesting  in  its  application  to  physical 
problems,  these  being  well  illustrated.  The  only  reason  for  including 
it  in  a  list  of  arithmetics  is  that  it  contains  some  work  on  proportion 
less  geometric  than  that  given  in  Euclid.    See  also  p.  193,  1539. 

Other  works  of  15^0.  Belli,  p.  343,  1573;  Boethius,  p.  27,  1488; 
Feliciano,  p.  148,  1526;  Forcadel,  p.  284,  1556-57;  Gemma,  p.  200, 
1540;  Glareanus,  p.  192,  1539;  Lonicerus,  p.  253,  1551  ;  Recorde, 
p.  214,  c.  1542  ;  Riese,  p.  139,  1522  ;  Tagliente,  p.  115,  15 15  ;  Anony- 
mous, '  Briefue  arithmetique  fort  facile  a  comprendre  '  ;  Johann  Weber, 
'  Gerechnet  Rechenbiichlein  auf  Erfurtischen  Wein-  und  Tranks-Kauff,' 
Erfurt,  with  a  second  edition  in  1583. 

Works  of  1571.  Digges,  p.  343,  1572;  Gemma,  p.  206,  1540; 
Riese,  p.  139,  1522;  Savonne,  p.  314,   1563;   Stifel,  p.  260,  1553; 


HIERONYMI 

CARDANI MEDIO 

LANBNSIS,    CIVISCIV'I-    B  O  N  O^ 

NIENSIS,     PH  ILO  S  O  P.ll  I.     MCDICI     ET 

Maihcniatici  ckirifsimi , 

OPVS  NOVVM  DE 

P  R  O  P  O  RT  I  O  N  1  B  V  S   N  V  M  13  R  O  R  V  M,  M  O 

V  VV  M,     P  O  N  D  E  P.  \'  ;.1 ,     S  O  N  O  R  V  M.     A  L  I  A  R  V  M  Q_V'  U    R  F:  R  V  At 

menfurajiclarumjnon  foliim  Gcomctrico  more  ftabiliium/cd  ctiatu 

uarrjs  cxocrimcntis  8(f  obfcruationibusrcrum  innatuia,rolcni 

dcmon(lranonci:lii(lratuiTi,acl  multipliccs  ufus  acs 

commodatum,&  in  Vlibros  digcflum. 

P   R   AE  T  E   R  E   A, 

ARTIS   MAGNA:,   SIVE  DE  REGVLIS 

ALGOBRAlClS,    LIBER    VNVS,     ABSTRVSISSIMVS 

dC  inexhauflus  plane  totius  Anthmetlca:  thefaurusjab 

authorcrccens  nuihis  tii  locis  rccogni- 

lusS^ai'dus, 

1  T  P.  M. 

DE    ALIZA    REGVLA    LIBBP.,    rtOC    EST,    ALGEBRAIC  AE 

logifticir  fiiafjiiumcros  recondita  numcrandi  fi'.btilitate.fcciindum  Gco^ 

metricas  quantitates  I'nquircp.ris ,  nccolliru  Coronis, 

nunc  deimtr.i  in  lucem  c  jita. 

Op;«  thyficii  cr  Mathematicis  Inprimis 
utile  Cf-ncceffaiinn. 


Cum  Ca^fMaicft,  Gratia  &  Priuilcgio. 

BASIL    E    >E. 

Fig.  165.   Title  page  of  the   1570  cakdax 


340  RARA   ARITHMETICA 

Trenchant,  p.  320,  1566;  Anonymous,  '  Les  principaux  fondemens 
d'arithm(§tique  '  ;  Nicolaus  Eschenburg,  '  Arithmetica  logiftica,'  Frank- 
fort ;  Alex.  Vandenbussche,  '  Arithmetique  militaire,'  Paris,  4°. 

FRANCISCUS  BAROCIUS.     Ed.  pr.  1572.       Venice,  1572. 

See  p.  295. 

Title.    See  Fig.  166. 

Description.  4°,  13.9  X  19.3  cm.,  the  text  being  10.3  x  17  cm. 
3  ff.  unnimib.  +  23  numb.  =  26  ff.,  17-36  11.    Venice,  1572. 

Edition.  There  was  no  other  edition.  A  German  translation 
was  published  in  Leipzig  in  16 16. 

This  is  an  attempt  to  popularize  the  mediaeval  number  game  of 
Rithmomachia  (Rithmimachia,  Rythmomachia),  set  forth  in  Latin  pos- 
sibly by  Shirwode  or  by  Faber  Stapulensis,  in  an  edition  of  Boethius  in 
1496  (see  p.  63),  and  afterwards  amplified  by  Claude  Boissiere  (see 
p.  271,  1556).  The  game  was  often,  with  no  authority,  attributed  to 
Pythagoras.  Barocius  (or  Barozzi)  amplified  the  treatment  attributed 
to  Faber  Stapulensis,  and  his  discussion  of  the  subject  is  clearer  than 
that  of  the  latter,  although  hardly  equal  to  that  of  Boissiere  already 
described.  He  had  already  published  a  philosophical  discussion  of 
arithmetic  as  stated  on  p.  295. 

LEONARD  AND  THOMAS  DIGGES. 

Ed.  pr.  1572.  London,  1579. 

Leonard  Digges  came  of  an  ancient  family  whose  seat  was  Digges  Court, 
Barham,  Kent.  He  studied  at  Oxford,  and  was  an  expert  mathematician  for 
the  time.    He  died  c.  1571. 

Thomas  was  a  son  of  Leonard,  and  was  born  in  Kent.  He  was  educated 
at  Oxford,  and  died  in  London,  August  24,  1595. 

Title.    See  Fig.  167. 

ColopJion.  '  Imprinted  at  Lon-/Alon,  by  Henrie  Bynneman, 
dwel-//ling  in  Thames  Street,  neere  vnto  //  Baynardes  Caftle.// 
Anno  1579.'     (P.  192.) 

Description.  4°,  12.8  x  18  cm.,  the  text  being  8.9  X  14.7  cm. 
16  pp.  unnumb.  +  191  numb.  =  207  pp.,  and  one  plan  (p.  176), 
35  11.    London,  1579. 

Editions.  London,  1572,  4°;  ib.,  i  579,  4°  (here  described) ; 
ib.,  1585,  4°;   ib.,    1590,  4°.    The   1579  edition  was  a  revision, 


IL  NOBILISSIMO 

ET  ANTIQVISSIMO 

GIVOCO  PYTHAGOREO 

NOMINATO 

Rychmomachia 
CIOE     BATTAGLIA 

DE     CONSONANTIE 

DE  NVMERI. 
Rilrouttle  jer  'vtilita,&  foW^  delli Studiop. 

Etal  prefente  per  Franccfco  BarozziGentil'huomo 

Venetiano  in  lingua  volgare  in  modo  di 

Paiaphrafi  compofto . 


IN  VENETIA. 
j^f>preJ[o  Cradofo  Pcrchacino,     157*' 


Fig.  166.    Title  pack  ov  bako^zi 


34- 


RARA   ARITHMETICA 


i*  An  Arichmeticall  Militare  Treati*(e,named 
STT^A^riOTICOS: 

Compendioufly  teaching  the  Science  of^J^ubers, 

as  well  in  Fracflions  as  Integers,  and  fo  much  of  the  Ru> 
Jesand  y£quations  AlgebraicalJand  Arte  of  Numbers 
Cofsicalljas  are  requifi'tefor  the  Profersionofa  Soldiour* 

Together  with  the  Modcrnc  Militare  Difcipline^OFficeSjLawcs  and 

Puctics  in  cucry  vvel  goucrncd  Campc  and  Annie  to  be  obferued ; 

Long  (Incc  attepted  by  L  b  o  n  a  a  d  D  i  o  g  e  s  Gentleman, 

Au2;mcntcd,di2:cftcd,  .ind  lately  [iniOicd,  by 


Thomas  D  i  g  g  e  s,  hii  Sonne. 


IVhei'cto  he  hath  alfoadtoyr.edccrtiiiire  QueTliofis  of  great  OrdinAmc^^ 

refeluedwhis  other  Treata^of  i^yr^techny  and  great 

Artilierie,  hereof ter  to  bee  publifhed. 

ViVET  Post  Fvnera  Virtvs. 


ayfT    LONDOKT: 

Printed  by  HenrieBynacman* 

jitfiic  Demini,  1 5  79  > 

Fig.  167.  Title  page  oe  the  1579  digges 


PRINTED    BOOKS  343 

for  the  preface  '  To  the  Reader '  (f.  a  2)  states  that  it  was  '  fin- 
ifhed  the  13.  of  October.  1579.' 

Considering  its  date,  this  work  is  a  very  good  introduction  to  the 
study  of  arithmetic.  The  arithmetic  proper  extends,  however,  only  to 
page  32.  Then  follows  a  brief  treatment  of  algebra  (pp.  33-51),  after 
which  are  certain  problems  (to  p.  70)  relating  to  military  matters. 
Pp.  81-19 1  are  devoted  entirely  to  miUtary  affairs. 

The  father  and  son  wrote  several  mathematical  treatises,  but  none 
directly  on  arithmetic.  One  was  the  '  Pantometria  '  of  1 5  7 1 ,  which 
De  Morgan  includes,  and  which  is  in  Mr.  Plimpton's  library,  but  which 
I  have  omitted  because  it  is  in  no  sense  an  arithmetic. 

Other  zvorks  of  1572.  Buckley,  p.  252,  1550  ;  Gemma,  p.  200,  1540  ; 
Grammateus,  p.  123, 1518;  Mariani,  p.  181, 1535  ;  Barlaamo,  '  Aoyio-riKT;, 
sive  arithmetic?e,  algebraicte  libri  VI  '  (in  a  work  on  spherics),  Strasburg, 
with  later  editions  at  Paris,  1594,  4°;  1599,  4°;  1600,  4°  (see  also  p. 
315,  1564)  ;  John  Seton  (see  Buckley,  p.  252,  c.  1550). 

SILVIO  BELLI.     Ed.  pr.  1573.  Venice,  1573. 

Born  at  Vincenza.     He  died  in  1575.     He  was  an  architect  at  Rome  and 
Ferrara,  and  wrote  on  practical  geometry. 

Title.    See  Fig.  168. 

Description.  4°,  15  X  20.6  cm.,  the  text  being  9  X  15.1  cm. 
46  ff.  (40  numb.),  19-21  11.    Venice,  1573. 

Editions.  This  is  the  only  separate  edition  of  this  semi-geo- 
metric work.  Belli  published  a  'Libro  del  misurar  con  la  vista,' 
which  passed  through  the  following  editions  :  Venice,  1565,  4°; 
ib.,  1566,4°;  1569,4°;  Venice,  1570,4°;  ib.,  1573,4°;  ib.,  1595, 
4°.  This  was  united  with  the  'Delia  Proportione'  in  1 595  (Venice) 
to  form  the  '  Quattro  libri  geometrici.' 

This  work  is  included  in  this  list  because  of  the  treatment  of  mediaeval 
proportion  which  it  contains. 

LUCAS  PAETUS.     Ed.  pr.  1573.  Venice,  1573. 

A  Venetian  jurist  of  the  sixteenth  century. 

Title.    See  Fig.  169. 

Description.  Fob,  20.5  X  29.5  cm.,  the  text  being  15  X  24.5 
cm.    8  pp.  unnumb  -\-  93  numb.  +  i  blank  =  102  pp.,  50  11.   Venice, 

1573- 


344 


RARA   ARITHMETICA 


SILVIO    BELLI 

V  I  C  E  N  T  I  N  O 

T>  E  L,LA 

PROPORTIONE,   ET   PROPORTIONALITA* 
Communi  Paflioni  del  Qjianto . 

L  I  B  RI     T  R  E. 

Z)uli,  (^  necejpirij  alia  ruera  ,  ^  Jkctle  intelligentia^ 

dell' Arithmetico-^ydella  Geometrio-^^O*  di 

tutte  le  Jcicntie  ^  arti. 

Al  Magnanimo  AlefTandro  Farnefo  Card. 


(^  0  N     T>  R  1  V  1  L  E  G  I  0. 

In  VtN eti a,  j^pprejfo  Francefco  de  rmncefchi Sanefc^,   1575, 


Fig.  168.   Title  page  of  belli 


PRINTKI)    HOOKS 


345 


LVCAE  PAETl 

I  VRISCONSVL 
DE  MENSVRIS.  ET 

PONDERIBVS    ROMANIS, 

ET    GRAECIS, 

cvM  HIS  <iyAE  HODiE  romae  svnt  collatis 

LIBRI     Q_V  I  N  QJ^E. 

BirSDEM    yjUtti^HyM  LECTIONVM    liber    VNiTS 

AD   SANCTISSIMVM    OPTIMVMCLVE    PRINCIPEM 
PIYM    C^VINCTVM     PONT.    MAX. 

MAXIMILIANl.il 


Fk;.  169.    Title  page  of  paetus 


346  RARA   ARITHMETICA 

Editions.  There  were  two  editions  of  this  work  pubUshed  at 
Venice  in  1573.    (See  next  title.) 

Ahhough  not  an  arithmetic,  this  work  is  a  scholarly  and  interesting 
contribution  to  the  history  of  the  weights  and  measures  of  Greece  and 
Rome,  and  the  symbols  inherited  by  the  Middle  Ages.  It  also  contains 
several  illustrations  of  ancient  measures. 

LUCAS  PARTUS.     Ed.  pr.  1573.  Venice,  1573. 

See  p.  343. 

This  is  a  different  edition  from  that  just  described.  The  title 
page  is,  however,  substantially  the  same. 

Description.  FoL,  17.5  X  23.2  cm.,  the  text  being  1 1.6  x  18.5 
cm.     144  pp.  (127  numb.),  38  11.    Venice,  1573. 

See  above. 

VALENTIN  MENHER.     Ed.  pr.  1573.  Antwerp,  1573. 

See  p.  249. 

Edited  by  Michiel  Cognet,  bom  c.  1549  at  Antwerp;  died  at  Antweip. 

Title.    See  Fig.  170. 

Colophon.    '  Antverpiae // Typis  Ant.  Dieft.  1573.' 

Description.  8°,  9.3  X  13.6  cm.,  the  text  being  6.^  x  11.8  cm. 
141  ff.  unnumb.,  24  11.    Antwerp,  1573. 

Editions.  I  have  no  doubt  there  was  an  earlier  edition,  although 
Cognet  may  have  edited  this  from  a  manuscript  left  by  Menher. 

This  is  one  of  the  best  of  the  purely  business  arithmetics  of  its  time. 
It  shows,  better  than  most  works  of  the  kind,  the  state  of  commerce  in 
the  second  half  of  the  sixteenth  century  in  Antwerp,  then  the  most  pro- 
gressive of  the  mercantile  cities  of  the  North.  In  it  may  be  studied  the 
merchandise,  the  trade  routes,  the  customs  of  merchants  and  bankers, 
and  the  prices  prevailing  in  that  period.  It  was  to  the  Netherlands 
what  Riese's  book  had  been  to  Germany  and  Borghi's  to  Italy.  As  the 
title  page  shows,  it  also  took  a  progressive  attitude  with  reference  to 
practical  geometry.     (On  Cognet  see  p.  365.) 

Other  works  of  iJYJ.  Kobel,  p.  102,  15 14;  Moya,  p.  310,  1562; 
Recorde,  p.  214,  c.  1542;  Peter  Beausard,  '  Arithmetices  praxis,' 
Louvain,  8°  ;  Simon  Kopfer,  '  Grundblichlein  der  Regel  Detri,'  Niirn- 
berg  ;  Bartolomeo  Piccini,  '  Trattato  de'  Cambi,'  Florence,  4°. 


PRINTED    BOOKS  347 

Works  0/  Jjy^.  Anonymous,  p.  244,  1546;  Baker,  p.  327,  15C8; 
Buckley,  p.  252,  c.  1550;  Jacob,  p.  298,  1560;  Riese,  p.  139,  1522; 
Rudolff,  p.  152,  1526  ;  Seton  (see  Buckley,  p.  252,  c.  1550)  ;  Lorenzo 


D'ARITHMETIQVE, 

conrcnant  pluficurs  belles  queftionsSc  de- 
mandcSjpropres  &  vtiles  a  tons  ccux  <jiii 

hantcntla  Trafiqucde  Marchanciifc. 
CorApoJe par  feU  Vaicmin'^^ennhcr^llcmand: 
rcuiU^corrige,&aiigmevte  enflujicurs  endrous 

parMichielCognet. 

ENSEMBLE 

Fne  am^lc  declaration  fur  le  fait  des  Changes » 

ITEM 

Vn  petit  difcours  dc  Lien  (jdeiiementdlfconter^diicc 
la  Solution  fur  diucrfes  opinions  y  propofees» 

AVEC 

La  Solution  des  cjneftions  Mathcmatiques 

parla  fiipputationde  Siniis^illuftrces  &am- 

piificcs  paries  demonftrations  Geome- 

triques  neceflaires  a  icelles. 


A    AN  VER  s 
Che:^  lean  waesherghe^ahfcu  de  Flandres, 

AVEC    PRIVILEGE, 

Fig,  170.    Title  page  of  the   1573  menher 

Bonocchio,  '  Breve  et  universale  risolutione  d'aritmetica,'  Brescia  (Ven- 
ice ?),  4°;  ib.,  1597  ;  Milles  de  Norry,  '  Arithm^ti(iue,'  Paris,  4°;  Johann 
Sekgerwitz,  'Rechenbiichlein  auff  allerley  Handthierung,'  Breslau. 


348  RARA   ARITHMETICA 

DIOPHANTUS.     Ed.  pr.  1575.  Toulouse,  1670. 

A  Greek  mathematician,  c.  300  a.d.     He  was  the  first  great  writer  upon 
algebra. 

Title.  '  Diophanti  //  Alexandrini  //  Arithmeticorvm  //  libri 
sex,//  et  de  nvmeris  mvltangvlis  //  liber  vnvs.//  Cvm  commen- 
tariis  C.  G.  Bacheti  V.C.//&  obferuationibus  D.  P.  de  Fermat 
senatoris  Tolofani  //  Acceffit  Doctrinae  Analyticae  inuentum 
nouum,  collectum  //ex  varijs  eiufdem  D.  de  Fermat  Epiftolis.// 
(Engraving  '  Rabault  Facit,'  with  motto  :  '  Obloqvitvr  nvmeris 
septem  discrimina  vocvm.')  Tolosae,  //  Excudebat  Bernardvs 
Rose,  e  Regione  Collegij  Societatis  Iefu.//M.  DC.  LXX.'   (P.  i.) 

Description.  FoL,  23.4  x  36.3  cm.,  the  text  being  15.1  x  23.9 
cm.  5  pp.  blank  +  6  unnumb.  +  341  numb.  +  48  of  notes  =  400 
pp.,  50-55  11.    Toulouse,  1670. 

Editions.  Basel,  1575.  There  was  no  other  sixteenth-century 
edition. 

Athough  entitled  an  arithmetic  this  is  really  a  treatise  on  algebra,  the 
first  systematic  one  ever  written.  It  contains,  however,  a  good  deal  of 
matter  upon  the  Greek  theory  of  numbers,  notably  the  '  Clavdii  Gasparis 
Bacheti  Sebusiani,  in  Diophantvm  Porismatvm,  Liber  Primus,'  '  Liber  Se- 
cundus,'  and  '  Liber  Tertius.'  A  certain  amount  of  this  work  also  enters  into 
the  treatise  itself,  but  this  is  generally  algebraic  in  character,  the  standard 
problem  requiring  the  finding  of  a  number  satisfying  given  conditions. 
This  leads  to  numerous  indeterminate  (Diophantine)  equations.  This 
edition,  by  Bachet  and  Fermat,  is  one  of  the  best  that  has  been  published. 

FRANCISCUS   MAUROLYCUS. 

Ed.  pr.  1575.  Venice,  1575. 

Francesco  Maurolico.  Born  at  Messina,  September  i6,  1494;  died  near 
there,  July  21,  1575.  He  entered  the  priesthood  and  later  became  professor 
of  mathematics  at  Messina.  He  wrote  chiefly  on  astronomy,  and  edited 
several  works  of  the  Greek  mathematicians. 

Title.    See  F^ig.  171. 

Colophon.  'In  monafterio  S.  Maria //a  parte  19.  lulij  die,// 
$.  II.  Indictionis,// 1553.'    (P.  305.) 

Description.  8°,  15.7  X  20.9  cm.,  the  text  being  10.6  X  17.5 
cm.  20  pp.  unnumb.  +  285  numb.-f  i  blank  =  306  pp.,  39-40 11. 
Venice,  1575. 


PRINTED    HOOKS 
D.    FRANCISCI 


349 


MAVROLYCL 

ABBATIS  MESSANBNSIS, 

Opufcula  Mathematica  •, 

y^^Qmc  frimti?n  tn  lucem  Aditn ,  cum  rerum  omnium 
notutH  dtgnamrru  - 

INDICE     LOCYPLETISSIMO. 

n?AGELLA  HFIC  PROXlME  CONTlG^A, 
CO  rum  Catalog  us  etl . 


CVM     PRIVILEGIO. 

Venetijsj  Aoud  FraiicifcumFiancircium  Senenfcm 
M       D      L     X     X    V. 


Kic. 


Title  page  of  the   Op  ii  sen  hi  of  maurolycus 


350 


RARA   ARITHMETICA 


Editions.  There  was  no  other  edition.  (See  below.) 
It  appears  from  the  colophon  that  this  work  was  composed  in  1553, 
although  it  was  not  published  until  1575.  It  includes  (pp.  26-47)  a 
'  Compvtvs  ecclesiasticvs  in  svmmam  collectvs.'  The  rest  of  the  treatise 
is  chiefly  astronomical.  It  forms  the  first  of  two  volumes  on  mathe- 
matics, the  second  being  the  '  Arithmeticorum  libro  duo  '  (see  below). 

FRANCISCUS  MAUROLYCUS. 

Ed.  pr.  1575.  Venice,  1575. 

See  p.  348. 

Title.    See  Fig.  172. 

Colophon.  '  Libri  fecundi  Arithmeticorum  Maurolyci  finis:  bora 
//decimaoctaua,  diei  Sabbati,  qui  fuit  lulij  249.  Cum  //  Meffanse 
cum  multo  pontis  &  arcus // apparatu  expectaretur  lo.  Cerda,// 
Methynenfium  Dux,//Prorex.  Indict.  15.//M.  D.  LVII.//  Vene- 
tiis,  M  D  LXXV.//Apud  Francifcum  Francifcium  Senenfem.' 
(R  183.) 

Description.  4°,  15.9  X  20.9  cm.,  the  text  being  13.3  X  17.2 
cm.     200  pp.  (175  numb.),  40  11.    Venice,  1575. 

Editions.  This  is  the  first  edition,  and  from  the  colophon  it 
appears  that  it  lay  in  manuscript  from  1557  to  1575.  A  second 
edition  appeared  in  Venice  in  1580.  The  'Arithmeticorum  libri 
dvo'  formed  the  second  volume  of  the  'Opuscula  Mathematica ' 
(Venice,  1575;  p.  348). 

The  work  is  mediaeval,  dealing  solely  with  the  Boethian  theory  of 
numbers.  It  was  one  of  the  last  of  the  extensive  sixteenth-century 
Italian  works  of  this  nature,  and  shows  considerable  originality  in  the 
treatment  of  figurate  numbers.  Maurolycus  was  by  no  means  a  mere 
compiler,  but  a  man  of  creative  power. 

HENRICUS  BRUCiEUS.     Ed.  pr.  1575.         Rostock,  1575. 

Born  at  Most,  Flanders,  c.  1531  ;  died  at  Rostock,  December  31,  1593. 
He  was  professor  of  mathematics  at  Rome,  and  later  professor  of  medicine 
at  Rostock. 

Title.    See  Fig.  173. 

Description.  8°,  9.5  X  15.1  cm.,  the  text  being  6.5  X  1 1.6  cm. 
76  ff.  (i  blank),  25  11.    Rostock,  1575. 


D.    FRANCISCl 

MAVROLYCL 

ABBATIS   MESSANENSIS. 
Mathematici  celcberrimi  > 

ARITHMETICOKVM  LlBRl  DVO, 

NVNC     PRIMVM     INLVCEM     EDITI. 

(um  rerum  omnmm  nomhtlmrru . 
INDICE    COPIOSISSIMO. 


CV  M     PR  I  VI  L  EGIO. 
Veneujs>  Apud  FrancifcumFrancifciuni  Sencnfcm. 
M      D      LX     X   V. 


Fig.  172.    Title  page  of  the  Lib ri  duo  o\-  maurolycus 


352  RARA   ARITHMETICA 

Editions.    There  was  no  other  edition. 

This  is  a  Latin-school  manual,  in  two  '  books.'  The  first  book  treats 
of  arithmetic,  and  chiefly  of  mediaeval  ratios.  The  second  treats  of 
algebra,  the  equations  being  considered  from  the  standpoint  of  geometry, 

HENRICI 

B  RV  Ci^I 

BELGiE, 

MATHEMaTICA1{VM 

exercitationv  m 

LIBRl   DVO. 


ROSTOCHII 

IKCVOEBAT      IftCOHVt      tVciVi 
TRANSYLVANVS. 

x^nno  M.  D.  LXXK 

Fig.  173.   Title  page  of  BRUc.tus 

and  some  attention  being  given  to  surd  numbers,  roots,  and  the  rule  of 
false. 

Other  works  of  1^75.    Gemma,  p.  206,  1540;   Kobel,  p.  102,  15  14; 
Lapazzaia,  p.  322,   1566;   Mariani,  p.   181,   1535;   Salignac,  p.  359, 


PRINTKI)    HOOKS  353 

1577:   Xylander,  p.  356,  1577:    Mauriciiis  SumiiiikjI/,  (icrsbach,  '  Arilh- 
metices  praecepta,'  Leipzig,  8°. 

Works  of  1576.  Gemma,  j).  200,  1540;  Petri,  j).  325,  1567  ;  'I'a- 
gliente,  p.  115,  15  15  ;  Joseph  Lange,  '  Arithmclica,'  Copenhagen,  S°. 

GIRJKA  GORLA  Z  GORLSSTEYNA. 

Ed.  pr.  1577.  Czcrny,  1577. 

A  Polish  Rechenmeister  of  the  huter  half  of  the  sixteenth  century. 

Title.    See  Fig.  174. 

Colophon.  'Wytifftenow  //  Starem  Hefte  Pra^  kem//  v  Girijka 
C.serneho.//Letha  Pane///M.  D.  LXXVII.'    (F.  XC,  r.) 

Dcsci-iption.  8°,  9.5  X  15.9  cm.,  the  text  being  7.2  x  1 1.7  cm. 
9  ff.  unnumb.  +  89  numb.  (Roman)  =  98  ff.,  21  11.    Czerny,  1577. 

Editions.    There  was  no  other  edition. 

This  is  one  of  the  few  arithmetics  in  the  Fohsh  language  published 
in  the  sixteenth  century,  and  is  very  rare.  There  was  a  copy  in  the 
Boncompagni  sale,  but  the  book  is  seldom  mentioned  by  bibliographers. 
It  consists  of  five  parts,  the  first  dealing  chiefly  with  the  fundamental 
operations  with  counters,  the  second  with  written  operations,  the 
third  with  fractions,  the  fourth  with  business  arithmetic,  and  the  fifth 
with  the  rule  of  false  and  allied  topics. 

DIONIS   GRAY.     Ed.  pr.   1577.  London,  1577. 

A  London  goldsmith  of  the  second  half  of  the  si.xteenth  century. 

Title.    See  Y\g.  175. 

Description.  Sm.  8°,  8.8  x  13.8  cm.,  the  text  being  6.5  x  12 
cm.     126  ff.  (8  unnumb.).    London,  1577. 

Editions.    London,  1577,  8°  (here  described);  ib.,  1586,  8°. 

This  is  a  practical  arithmetic,  consisting  of  four  parts.  '  The  firft 
containeth  foundrie  partes  of  Arithmetique,  that  is  to  fay,'  the  funda- 
mental operations  with  integers,  including  progressions.  '  I'he  feconde 
parte,  containeth  the  faid  partes  feruyng  for  practife  of  broken  numbers 
or  fraccions.'  '  The  third  part  containeth  the  fondrie  Rules  of  pro- 
portion, furthered  by  vfe  of  the  forefaid  partes,'  and  includes  alligation 
and  the  rule  of  false.  '  The  fourth  parte  containeth  fondrie  Rules  of 
breuetie,'  or  short  processes.  It  is  one  of  the  earliest  English  arith- 
metics to  contain   rules  and  definitions  in  rhyme.      For  example,  in 


354 


KARA   ARITHMETICA 


M.    D.    LXXVIL 


1^ 


-'J 


».  -^-- 


Fic.  174.   Title  page  of  girjka  gorla  z  gorlssteyna 


PRINTHl)    HOOKS 


355 


speaking  of  addition  (".ray  says 
here  a  fewe  lines  in  verie  : 


And  for  lo  amplilie  the  clfecle,  lake 


^  i^Theftore-hoiifeof  W. 

Breuitie  in  <-v'-uoorkes  of\A-  f.  v 

rithcmctikc  ,  containyng  afvvcll  the  ^r^ 

foHndrie  fanes  of  the  Science  tn  m  hole  y^ 

anDb?oUcn  numbcr0,  Uiicb  cljc  V^w-.  tt 

lc0  of p?opouton,funfji  rcu  to  p^oS^  i§ 

table ufc :  3s  alfo  fiinuerie  rules  of  W 

a5?emticofU)ojhe,ofrdrc,plea--  Sj 

faunte.  anD  commoDiouisf  c.y 

c{fcctr,fecfo?tlKbp  KK 

"DioyjisCrayQf  Vi 

lonbon  W 

CoiDfmitf;.  fc 

1577.  ^ 

fflmpriKtcd  at  LZdonfor  jVtlUam  h^ 

Narto-^yVidlhon  Har.-fo.'7y  >X).- 

dv:ellyng'tn  l^aulcs  \  ■^ 

Church-yard.  ^' 


Fig. 


75- 


.K    PAGE    OK    GRAY 


'  C.0£  fondrie  fommes  perticulars,  one  totall  for  to  frame, 

Set  them  doune  right  orderly,  as  worke  doeth  beft  require  : 
What  place  ye  giue  to  any  one,  the  reft  let  haue  the  fame. 
So  maie  you  well  performe  the  'ffecte,  of  what  you  doe  defire. 

The  rule  is  then  continued  in  a  series  of  verses. 


356  KARA   ARITHMETICA 

GUILIELMUS  XYLANDER. 

Ed.  pr.  1577.  Heidelberg,  1577. 

WiLHELM  HoLTZMANN.  Boin  at  Augsbuig,  December  26,  1532  ;  died  at 
Heidelberg,  February  10,  1576.    He  was  professor  of  Greek  at  Heidelberg. 

Title.    See  Fig.  176. 

Colophon.  '  Excudebat  lacobus  Mylius,  impenfis  //  Matthaej 
Harnifch.//M.  D.  LXXVII.'     (F.  52,  v.) 

Description.  4°,  14.9  X  19.2  cm.,  the  text  being  10.5  X  16.  i 
cm,     2  ff.  unnumb. +  50  numb.  =  52  ff.,  31  11.    Heidelberg,  1577. 

Editions.  There  was  no  other  edition  of  this  work.  Xylander 
translated  several  Greek  works  into  Latin,  among  them  Euclid 
(Basel,  1562)  and  Diophantus  (Basel,  1575,  fol).  The  'Opuscula' 
appeared  a  year  after  his  death.  Xylander  also  edited  Psellus 
(Basel,  1556). 

As  the  title  states,  this  work  is  divided  into  four  parts.  The  first 
and  most  extensive  (ff.  1-22)  relates  to  astronomy.  The  second  is 
purely  arithmetical  (ff.  22-36),  and  treats  of  common  fractions,  giving 
the  usual  operations  and  the  rule  of  three.  The  arrangement  of  this 
part  is  peculiar,  addition  and  subtraction  being  treated  together,  after 
which  division  is  explained,  multiplication  coming  last.  This  order 
would  be  justified  if  Xylander  had  reduced  his  fractions  to  fractions 
having  a  common  denominator,  but  he  merely  follows  the  usual  rule  of 
cross  multiplication  (f.  29,  v.).  The  third  section  is  '  De  svrdis,  qvos 
vocant,  nvmeriis  iis,  qvi  a  qvadratis  primo  nafcuntur,  Inftitutio  do- 
cendo  explicanda,'  a  chapter  now  conventionally  placed  in  our  alge- 
bras. The  fourth  section  relates  to  the  celestial  globe  and  the  astrolabe 
(ff.  46-50). 

PIETRO  ANTONIO  CATTALDI. 

Ed.  pr.  1577.  Bologna,  1577. 

Cataldi,  Cataldo.  Born  at  Bologna  in  1548  ;  died  at  Bologna,  February 
1 1,  1626.  Professor  of  mathematics  and  a.stronomy  at  Florence  (1563),  Peru- 
gia {1572),  and  Bologna  (1584).  He  wrote  several  mathematical  works,  and 
to  him  is  due  the  beginning  of  the  theory  of  continued  fractions  (1613). 

Title.    See  Fig.  177. 

Description.  4°,  14.3  X  19.6  cm.,  the  text  being  10.9  X  16.7 
cm.    8  ff.  unnumb.,  40  11.     Bologna,  1577. 


PRIN'l^:i)    I'.OOKS 


357 


OP VSCVL A 

MATHEMATIC  A 

DOMINI 

AVGVSTAKL 

tyi/fhrifmi  CofmogntphicUiierL 

t)e  Surdorum  Numerorum  natura  &  tmUatme  Uheu 
Ve  yfu  Glohi  &  TUnis^harij  era^dtut. 


ffKrn. 


HE  I  D  E  L3  E  j^  G  j£ 

Excudcbatlacobus  Mylius ,  impenfis 
MatthaeiHarnilch. 


Fig.  176.   Title  page  of  xylander 


358  RARA   ARITHMETICA 

DVE     LETTIONI 

DI     PIETR ANTONIO 

CATTALDI    BOLOGNESE 

fATTE  NELL/ICADEMIA 
dei  Dt[jegno  dt  Pemgta, 

ALLI  GENEROSI,ET  VIIITVOSISSIMI 

Signori  Academici ,  il  Signer  Caualier  Paciotto, 
&  il  Signor  Caualieio  Anallagi  • 


In  Bologna. 
PerGiouanni  Rofsi  mdlxxvi  f. 

Con  Ikentia  de  Supertcri , 

Fig.  177.    Title  page  of  cattaldi 


PRINTED    BOOKS  359 

Editiois.  There  was  no  other  edition.  Cattalcli  wrote,  under 
the  pseudonym  Pcrito  Annotio,  a  '  Prima  Parte  dclla  Pratica 
Aritmetica '  (Bologna,  1602)  and  a  '  Trattato  dei  numeri  per- 
fetti '  (Bologna,  1603),  but  like  all  of  his  strictly  mathemati- 
cal works  they  appeared  after  1600.  The  second  part  of  the 
'Pratica'  appeared  under  his  own  name  in  1606. 

This  book  hardly  deserves  to  be  ranked  in  a  list  of  arithmetics.  I 
have  included  it,  however,  because  in  his  first  address  Cattaldi  treats 
somewhat  of  numbers,  and  in  the  second  address  he  applies  arithmetic  to 
mensuration.  The  treatment  is  mediaeval,  and  the  chief  interest  in  the 
book  is  a  typographical  one  ;  for,  the  printer  not  being  able  to  set  such 
numbers,  all  of  the  fractions  have  been  written  in  by  hand. 

Other  uwrks  of  i^yj.  Borghi,  p.  16,  1484  ;  Buckley,  p.  252,  c.  1550  ; 
Capella,  p.  68,  1499  \  Cressfelt,  p.  290,  1557  ;  Herbestus,  p.  303,  1561  ; 
Hobel,  p.  314,  1563  ;  Ramus,  p.  331,  1569  ;  Seton,  p.  252  (see  Buckley, 
c.  1552);  Anonymous,  *  Arithmetica,'  Debreczin  ;  Miguel  Berenguer, '  De 
numerorum  antiquorum  notis,'  Saragossa ;  (ieorg  Gehrl,  '  Ein  nutzlich 
and  kiinftlich  Rechenbuch  auff  der  Federn,'  Prag,  8° ;  Johann  Jung,  an 
arithmetic,  Liibeck,  c.  1557  (no  copy  extant?);  Bernhard  Salignacus, 
'  Tractatus  de  Arithmetica  Partium  et  Alligationis,'  Frankfort  (Peacock 
says  1575);  also  '  Regula  veri,'  Heidelberg,  1578,  and  '  Arithmeticae 
Libri  II,  et  Algebrae  totidem,'  Frankfort,  1580,  4°;  ib.,  1593,  4°- 

JACQUES  CHAUVET  CHAMPENOIS. 

Ed.  pr.  1578.  Paris,  1578. 

Professor  of  mathematics  in  the  University  of  Paris,  in  the  second  half  of 
the  sixteenth  century. 

Title.    See  Fig.  178. 

Description.  8°,  10.  i  X  17  cm.,  the  text  being  7.5  X  14  cm. 
9  pp.  unnumb.  +  383  numb.  =  392  pp.,  28-29  11.     Paris,  1578. 

Editions.  There  was  no  other  edition.  The  privilege  is  dated 
'  a  Paris  le  huictiefme  iour  de  Septembre,  M.  D.  LXXVII.',  and 
the  dedicatory  epistle  '  De  Paris  ce  28.  de  Nouembre  1577.' 

The  chief  interest  in  the  book  lies  in  the  fact  that  it  was  written  by 
a  man  who  was  so  interested  in  military  matters  as  to  take  a  large  num- 
ber of  his  applied  problems  from  army  life.  The  arrangement  of  the 
book  is  not  peculiar,  but  the  great  array  of  military  problems  is  unique, 
and,  having  been    j)rei)ared  especially   for  this  work,  would  form  an 


36o  RARA   ARITHMETICA 

LES 

INSTITVTIONS 

DEL'ARITHMETIQVEDB 

I  A  C  C^V  ES     CHAVVET    CHAMPE- 

nois,ProfelIeUr  es  Mathcmatiques  ea 
rVniuerfite  de  Parisjdiuifecs  en 
quatre  parties.-auec  vn  pe- 
tit Traide  des  fradlions 
Aikonomiques. 


A    PARIS, 

CheT.  Hierofme  de  Marnef,au  mont 
S.Hilaire,arenfeigne  du  Pelicaric 

1578. 

AVHC    PRIVILEGE    DV     ROY.. 

Fig,  178.   Title  page  of  chauvet  champenois 


PRIN'ri^)    IU)()KS  361 

interesting  source   for  the  study  of  army  conditions  in   1^'rance  in  the 
sixteenth  century. 

Other  works  of  i^jS.  Capella,  ]).  66,  1499  ;  Oemnia,  \).  200,  1540  ; 
Ramus,  p.  355,  1569;  SaHgnacus,  \).  359,  1577;  'Jarlagha,  p.  278, 
1556  ;  Trenchant,  p.  320,  1566. 

JOHANN  OTTO.     Ed.  pr.  1579.  Leiivjg,  1579. 

A  Freiburg  Rechenmeister,  born  c.  1529.  The  dedicatory  epistle  is  dated 
1 579,  and  is  .signed  '  Johan  Otto.  yEtatis  f  ua;  50,'  which  approximately  fixes 
the  date  of  Otto'.s  birth. 

Title.    See  Fig.  179. 

Co/ophon.  '  Getruckt  zu  Leipzig/  bey  Johan  //  Rhambawcs 
Seligcrs  hinderlaf-//nen  Erben/  I579-'     (?•  423-) 

Description.  4°,  14.5  X  19. i  cm.,  the  text  being  11.7X  17.5 
cm.     424  pp.  (381  numb.),  28-38  11.     Leipzig,  1579. 

Editions.    There  was  no  other  edition. 

The  book  is  composed  almost  entirely  of  tables,  although  the  first 
few  pages  give  a  brief  treatment  of  counter  reckoning. 

CHRISTIAN   URSTISIUS.     Ed.  pr.  1579.  Basel,  1579. 

Allassiderus,  Allassisiderus,  Wursteisen,  Urstis.  Born  at  Basel  in 
1544 ;  died  at  Basel  March  30,  1588.  He  was  educated  at  Basel  and  became 
professor  of  mathematics  (1565)  and  aftei-wards  {1585)  of  theology  in  that 
university. 

Title.    See  Fig.  180. 

Colophon.  '  Basileae  Helve-//tiorvm,//per  Sebastianvm  Hen- 
//ricpetri,  An.  hvmanitatis // Filii  Dei,  CI3.  ID.  LXXIX.// 
Menfe  Augufto.'    (P.  192.) 

Description.  8°,  10.3  X  16.6  cm.,  the  text  being  6.4  X  i  1.2  cm. 
192  pp.,  25-28  11.    Basel,  1579. 

Editions.  On  p.  192  is  a  woodctit  with  the  date  1569,  evi- 
dently used  from  some  earlier  work  of  the  printer's.  Murhard 
(vol.  I,  p.  173)  mentions  a  work  by  Urstisius,  '  Zwey  Biicher  von 
der  Rechenkunft,  defsgleichen  in  der  deutfchen  Sprache  nie  aus- 
gegangen,'  Basel,  1569,  4°,  in  which  this  woodcut  may  have 
been  used.  He  also  mentions  an  edition  at  Basel  in  1595.  An 
English  translation  by  T.  Hood  appeared  in  1596. 


Calculator. 

bttc^  fut^dllc/  foArithmeticam  ItCb  ^(ttett/ 
3tifbttt)cr^rit  a6rr  ffirKaufRcut  /  2fmpt$perfoitctt/ 

J:34nD(cr/j^amatJttDalIefampt  (^iefden5fcoDafau|5(cJn^ffc6< 

aaff  We  aa?as:t»n^©ewf<^ob^dt^terKt>tic^m  ^5d  etoot  itdpiiflrgetf 
(left:  2ri!cljanC>ctn0rfbtm  gW^ofrtrofctcnflUcfe/otd  iu  5rancf  fort  art  dcr 
C6cr/  ^u  5f  f  i&er0  /  ^rcf^w/  2vmiptni^  ^n^  2fittt«kr0  /  etc-  vnb  in  fumma 
aHcntfjalbctt/  tt>o()fr(rcnf«crauff  no.  ffun5mw0etcUrt<(l.  X>atimmaVU 
f  cuff  miff  (Tcntiier  gut/  »n&  anborcr  fantcmfcflw  tv^arm  attdtta^cnfef  eums 
ma  gar  be()en6  on?  Muliipliciren  t»n^  Diui<liren ,  alWtt  tttrt^  bkaUorWc^ttPc 
fpecies  ter  Addition  ob^reumtnirettd/  Vit^ottc^  andttemHatceUc^^ttiu 
fccrj  tattfoit  ExcmplaVtt55ra0m^ttpn&w/im&  atttfittiafm  fdii-  SQ^od 
a&frmf&r<rdittM^r{m;Crttrt0«O?rtflf/  iftrtuff^ 

fiittftcStudiofam^^tt  ^rdBrr^itt  ^erattm 

Pfalm.90. 

iMandmiVOMlKVS  angeiisjuisdeu^'vifupfidmt 
U  m  ommbus  vjjs  tuis^ 


^dpUsI 


Fin.  179.   Title  page  of  otto 


PRINTED   BOOKS 


363 


This  is  a  book  written  by  a  gymnasium  teacher,  who  was  filled  with  a 
love  of  the  classical  learning,  and  yet  who  recognized  that  the  old 
Boethian  arithmetic  must  give  way  to  the  practical  treatment  demanded 

ELEMENTA 

A  R  I  T  H- 

METlCiE,    LOGU 

CIS.   L  E  G  IB  VS 

DED  VC  T  A, 

In  ufum  Academiae  Bafil, 
'  Opera  Grfiudio 

Christian!    Vrstisit, 

Mdthemat'icdrMri  profffi^ 

forls. 


BASILED, 
PER.  Sebastianvm  Hen- 
ri CPETR.U 

Fig.  180.   Title  i-acik  of  urstisius 

by  modern  conditions.  As  a  result  it  is  a  somewhat  heavy  treatment  of 
part  of  the  old  theory  of  numbers,  together  with  the  fundamental  oper- 
ations, roots,  and  certain  commercial  applications  of  the  rule  of  three, 
of  partnership,  and  of  alligation.    Urstisius  expresses  his  indebtedness 


364  RARA   ARITHMETICA 

chiefly  to  Euclid,  Ramus,  Salignacus,  Gemma  Frisius,  and  Scheubel,  not 
a  very  extensive  list  for  1579. 

Other  works  of  1579.  Digges,  p.  340,  1572 ;  Gemma,  p.  200,  1540; 
Mariani,  p.  181,  1535;  Riese,  p.  139,  1522;  Tagliente,  p.  115,  15 15; 
Anonymous,  '  Nouvelle  et  facile  methode  d'arithm^tique,'  Lyons,  16°; 
Miguel  de  Eleyzalde,  '  Guia  de  contadores,'  Madrid,  4°. 

GIAMBATTISTA  BENEDETTI. 

Ed.  pr.  1580.  Turin,  1585. 

Johannes  Baptista  Benedictus.  Bom  at  Venice,  August  14,  1530;  died 
at  Turin,  January  20,  1590.  Philosopher  and  mathematician  of  the  Duke  of 
Savoy. 

Title.  '  lo.  Baptistae  //  Benedicti  //  Patritij  Veneti  Philofophi. 
//Diversarvm  Specvlationvm  //  Mathematicarum,  &  Phyficarum 
//  Liber.// Quarum  feriem  fequens  pagina  indicabit.// Ad  Sere- 
nissimvm  Carolvm  Emanvelem  //  Allobrogvm,  et  Svbalpinorvm 
//DvcemInvicti.ssimvm.//(Large woodcut.)  Tavrini,ApudHaere- 
demNicolaiBeuilaqu3e,MDLXXXV.//Superioribuspermiffum.' 

(F.  3-) 

Description.  Fol.,  20.7  X  30  cm.,  the  text  being  14.5  X  23  cm. 
425  pp.  numb.  +  10  unnumb.  +  i  blank  =  436  pp.  (118  in  the 
part  devoted  to  arithmetic),  42-46  11.    Turin,  1585. 

Editions.  Turin,  1580,  fol.;  ib.,  1585,  fol.  (here  described); 
Venice,  1599,  fol. 

This  work  is  composed  of  six  parts,  of  which  the  first  is  entitled 
'  Theoremata  Arithmetical  The  other  five  parts  relate  respectively  to 
perspective,  mechanics,  Aristotle,  Euclid's  book  on  proportion,  and 
physics.  The  arithmetic  is  a  scientific  consideration  of  various  matters 
of  theory,  and  is  best  illustrated  by  the  following  theorems,  which,  as  is 
usually  the  case,  are  stated  in  the  form  of  questions  :  'Theorema  IIII.  Cvr 
multiplicaturi  fractos  cum  integris,  recte  multiplicent  humerantem 
fracti  per  numerum  integrorum,  partianturque  productum  per  denomi- 
nante  fracti,  ex  quo  numerus  qusefitus  colligitur.'  'Theorema  XXIX. 
Qvid  caufffi  eft,  cur  fubtracto  duplo  product!  duorum  numerorum  ad 
inuicem  multiplicatoru  ex  fumma  fuorum  quadratorum,  Temper  quod 
fuper  eft  duorum  numerorum  quadratum  differentiae  fit?  '  All  such 
questions  are  answered  by  the  aid  of  diagrams,  quite  as  Euclid  would 
have  done.  The  graphic  treatment  is  even  applied  to  such  problems  as 
that  of  the  couriers  (/Theorema  CXIIII').    There  is  an  interesting 


PRINTr:i)    HOOKS  365 

'  ApiK'mlix  (Ic  spccvlalione  regvlac  falsi,'  which  close-s  the  aiilhniclK  , 
and  sets  forth  an  elaborate  explanation,  with  grajihit  aids,  of  the  nde  of 
false  which  was  then  so  common. 

As  a  specimen  of  graphic  arithmetic,  combining  the  ICuclidean  the- 
ory with  the  Renaissance  practice,  Benedetti's  book  is  worthy  of  more 
attention  than  it  has  received.  It  may  be  inferred  from  some  of  his 
statements  that,  although  purely  a  theorist  himself,  he  recognized  the 
obsolete  nature  of  much  that  the  practical  arithmetics  had  to  offer. 
It  is  possible  that  the  common  partnership  problems  were  already  con- 
sidered too  traditional,  for  he  says  :  '  Svpponunt  antiqui  aliquot  merca- 
tores  dantes  pccunias  lucro  in  diuerfis  vnius  anni  tcmporibus,'  etc. 

ISAAC  RIESE.     Ed.  pr.  1580.  Leipzig,  1580. 

One  of  the  five  sons  of  Adam  Kiese  (p.  138). 

Title.    See  Fig.  i8i. 

Colophon.  'Gedruckt  zu  Leipzig  //  ben  Mans  Rhambaw/  //im 
Jar//M.  D.  LXXX.'     (P.  402.) 

Description.  4°,  14.  i  X  17.9  cm.,  the  text  being  11  x  16.5  cm. 
36  pp.  unmimb.  +  366  numb.  =402  pp.     Leipzig,  1580. 

Editions.    There  was  no  other  sixteenth-century  edition. 

Although  this  book  contains  a  brief  treatment  of  the  operations,  it 
is  largely  made  up  of  tables  for  the  use  of  merchants.  It  is  therefore 
not  a  textbook,  and  1  believe  it  went  through  only  one  other  edition 
(Leipzig,  1 61 9). 

WILLEM   RAETS.     Ed.  pr.  1580.  Antwerp,  1580. 

A  Dutch  arithmetician  of  Maastricht.  The  privilege  is  dated  May  22,  1576, 
and  mentions  only  the  name  of  Raets.  Coignet  (Cognet)  in  his  preface,  how- 
ever, speaks  of  his  particular  friend  ('mijn  zoderiinge  goet  vrient ')  Raets  as 
dead  ('meynen  ouerlede  vrient '),  so  that  he  very  Hlcely  died  between  i  576  and 
1580.    Coignet  was  also  the  editor  of  Menher's  arithmetic  (see  p.  346). 

Title.    See  Fig.  182. 

Description.  8°,  9.8  X  14.7  cm.,  the  text  being  7.4  X  i  1.9  cm. 
88  ff.  unnumb.,  27-31  11.     Antwerp,  1580. 

Editions.    There  was  no  other  edition. 

This  work  is  one  of  several  elementary  business  arithmetics  ap]:)earing 
in  the  Low  Countries  about  this  time.  It  is  a  small  book  of  no  special 
merit  save  as  it  shows  the  style  of  commercial  problems  of  the  period. 
The  similarity  of  the  title  page  to  that  of  Menher's  'Arithmetiqve 
seconde'  of  1556  (p.  283)  is  interesting. 


366 


RARA  ARITHMETICA 


5tS^ 


S 


farii  o.^^f^nt)tgctt)ir^tgct/  t^arinticn  Die  ^c^alutige 
in  (inf  auffcn  mt)  t?crf  auffcn/  one  fonDctc  irciilc  ufftiac  ^cc^nung 
^arbf ^enD  ^u  ftntcn :  Zmfi  t>on  aUctlc 9  a)?a^  /  dUn  I  t>nt)  ® c*» 
wfc^t  fauff.  ©amp(  me^r  angc^cngtcn  £a jfc(n  auff  t)<c  C5)?a=« 

©(ttjfc^f  gcrtc^tef.  ixeslctc^m  cine  ^cc^fel  $)^cc^nun3  auff 

^In^^nb® 0101/1?.  ^m&aufflcutm/  ^cntJlcrn/ 

:^potC(f cm/  ©c^#rn/ 1?.  O^tit)  Dcm  9cm<i* 

nm  0)?<»nn  fc^r  Df  cnfltic^  a»3^^ 

^(b^mfejlci0att|ft>ieSJWWcf;eSyfiJnftctnib 
au  £cipMg/  "SoriiMt^  Der  ^cfwrt  in 

£S)?lt€Wilt|IHc^ere«^firc^«Q5c3ttaMtti3t)nt> 


^ 


Fig.   181.   Title  page  of  isaac  riese 


PRINTKD    BOOKS 


367 


Other  ivorks  0/  ij8o.  Apiamis,  p.  155,  1527;  Baker,  p.  327,  15C8  ; 
Cassiodorus,  p.  211,  1540  ;  (lemma,  p.  200,  1540;  Gyraldus,  p.  254, 
1552  ;  Mariani,  p.  180,  1535  ;  Maurolycus,  p.  35o.  i575;  Ramus,  p.  331, 
1569;   Ringhieri,  p.  253,  1551;  Salignacus,  p.  359,  1577;  J.  Ammo- 


.JpnH^IMETICA 

^^  Oft 

etn  nteib  Ciiffetboecft/ftaii 

mikm  aaets/  iHE^acftncljtec. 

7Va€r  in  die  Fondamenten  feer  grondelijck  verclaert 
eR  met  vccl  fchoone  qucflien  ghcilluftreert  wor- 
den,tot  nut  ende  oorbaer  van  alle  CoopUede 
ende  licfhcbbers  der  feluer  Conftcn. 

Met  noch  een  Tra£laet  vande  VViJfclroede,  met  Amo^ 
fatten  verc'tert,  door  Mich'iel  Coimi 


T'hantwerpen, 

Ztn  bupfe  tjan  1^enD?ich  l^cnUucreu/  intrc 
tdicbiocme.  j    <>    8  o. 

Met  Priuilcgic  van  thicn  lacren. 

Fig.   182.   Title  page  of  raets 


nius,  '  Isagoge  Arithmetica,'  Wittenberg,  8  °;  H.  Flicker,  '  Arithmetices 
introductio,'  Cologne,  8°,  and  '  Compendium  calculorum,  seu  projectili- 
um  ratiocinationis,'  ib.,  8°;  C.  Zuccantini,  '  Libro  d'Albaco,'  Siena,  12°. 


368  KARA   ARITHMETICA 

ANTONIO  MARIA  VISCONTI. 

Ed.  pr.  I  581.  Brescia,  1581. 

A  mathematician  of  Piacenza,  of  the   latter  part  of  the  sixteenth  century. 

Title.    See  Fig.  183. 

Colophon.  '  Brixiae.//  apud  lacob.  &  Policretum  de  Turlinis.// 
M  D  L  XXXI.//Svperiorvm  Premissv.'     (P.  301.) 

Description.  4°,  14.8  X  20.2  cm.,  the  text  being  9.4  X  15.6 
cm.     304  pp.  (289  numb.),  37  11.     Brescia,  1581. 

Editions.  There  was  no  other  edition.  Riccardi  believes  the 
date  155  I  in  Murhard  is  a  misprint.     I  find  no  such  edition. 

This  rare  and  curious  book  is  a  combination  of  algebra,  advanced 
arithmetic,  geometry,  and  the  mensuration  of  river  lands.  The  arith- 
metic is  designed  to  be  an  application  of  the  algebra,  and  includes 
roots,  equation  of  payments,  proportion,  and  a  little  bookkeeping. 

Other  works  of  1^81.  Anonymous,  p.  195,  1539  ;  Fischer  (Piscator), 
p.  247,  1549;  Gemma,  p.  200,  1540;  Lonicerus,  p.  253,  1551; 
Peverone,  p.  290,  1558;  Ramus,  p.  330,  1569;  Riese,  p.  139,  1522  ; 
Johann  Kandleon,  '  Arithmetica,'  Regensburg  (referred  to  in  the  care- 
lessly prepared  Boncompagni  sale  catalogue,  but  probably  the  Kaudler 
book  of  1 591).  There  was  also  'A  short  Introduction  to  Arithmetic' 
published  anonymously  in  London  c.  1581-90,  8°. 

JULIUS  CAESAR  of  Padua. 

Ed.  pr.  1582.  Frankfort,  1678. 

A  German-Italian  teacher  of  the  sixteenth  century. 

Title.  '  Julii  Caesaris  //  von  Padua  //  Arithmetifche  //  Prac- 
tick/  //  Welche  in  alien  Landern  //  fehr  nutzlich  kan  gebraucht 
//  werden/  bey  Kauff-  und  Ver-//kauffung  allerley  Wahren/ 
auch//die  grotlen  Muntz-Sorten  in //kleine/und  die  kleine  in 
groffe  //  zu  verwandeln ://  Samt  der  Erklarung/  wor-//inn  ein 
Jedweder/der  nur  die // Ziffern  kenet/alfobald  fehen  kan/// 
wie  diO  Buchlein  zu  verftehen  ift.//  Nebenft  Morgen-  und  Abend- 
//Gebehtenund  Gefangen/den//reyfenden  Perfonen  gar  bequem 
//  bey  fich  zu  fuhren.//  Und  dann  letzlich/  eine  kurtze  Be-// 
fchreibung/  der  denckwurdigften  //  Sachen/  fo  von  Anfang  der 
Welt/ biO // zu  diefer  Zeit/ vorgangen.// Franckfurt  am  Mayn/ 
//Druckts  Blafius  Jll^ner/im  Jahr  1678.'     (P.  i.) 


PRINTED    BOOKS 


369 


ANTONII    MARIAE 

V    I    C    E   C   O   M   I   T    I   S 

CIVIS    PLACENTINr, 

Pradica Numerorum,  &  Menfurarum,  ac  Alluuionispatticioneni, 
iaueftigandi ,  &  vc  in  Indice  fcquenti . 


B  R  1  X  1  AE 


APVD    lACOBVM,    ET    POLYCK£TVM 
dcTurlinisFratrcs.         158*' 

Fig.   181.    Title  pace  ok  \  isconti 


370  KARA   ARITHMETICA 

Description.  12°,  4.7  X  11  cm.,  the  text  being  3.9  x  9.8  cm. 
504  pp.  (239  and  258  numb.)  ;  tables  28  11.,  other  pages  22  11. 
Frankfort,  1678. 

Editions.  Strasburg,  1582,  16°;  ib.,  1583;  ib.,  1585;  ib., 
1592,  and  after  1600  as  late  as  1679. 

The  first  part  of  the  work  is  devoted  to  multiplication  tables.  This 
is  followed  by  a  chapter  on  chronology.  The  last  part  of  this  edition 
is  a  separate  book  of  prayer, '  Chriftliche  Morgen-  Und  Abend  Gebeht,' 
of  the  same  date  (1678).  There  is  nothing  arithmetical  in  the  work 
except  the  tables. 

GASPARO  SCARUFFI.     Ed.  pr.  1582.  Reggie,  1582. 

An  Italian  jurist  of  the  sixteenth  century. 

Title.    See  Fig.  184. 

Colophon.  'In  Reggie,// Per  Hercoliano  Bartoli. //  M.D.- 
LXXXIL'     (F.  65,  V.) 

Description.  FoL,  21  X  30.2  cm.,  the  text  being  12.7  x  19.5 
cm.    65  ff.,  32  11.    Reggie,  1582. 

Editions.  This  is  the  only  sixteenth-century  edition  of  Sca- 
ruffi's  work,  the  privilege  being  dated  July  15,  1582.  The  dedi- 
catory epistle  is,  however,  dated  at  Reggie,  May  16,  1579. 

The  work  is  called  in  the  running  headlines  a  '  Discorso  sopra  le 
monete,'  and  is  a  historical  treatise  on  money  and  coinage,  touching 
slightly  on  exchange. 

GASPARO  SCARUFFI.     Ed.  pr. '1582.  Reggie,  1582. 

See  above. 

Title.  '  Breve  Instrvttione  //  sopra  il  discorso  //  f atto  dal  Mag. 
M.// Gaspare  Scarvffi,// per  regelare  le  cese  delli //danari.// 
(Woodcut  representing  a  bishop,  surrounded  by  these  words:  S. 
Prosper  //  Episcopvs  //  Regii  //.)  In  Reggie,//  per  Hercoliano 
Bartoli.// M.  D.  LXXXII.'    (F.  i,  r.) 

Colophon.  'Di  Reggie  il  xvij.  Aprile.  M.D.LXXXI.'  (F. 
9,  r.)     The  colophon  and  title  page  do  not  agree  as  to  date. 

Description.  FoL,  21  X  30.2  cm.,  the  text  being  12.8  x  20  cm. 
9  ff.,  27-30  11.    Bound  with  the  preceding  work.    Reggie,  1582. 


PRINTKl)    BOOKS 


DIM.GASPARO  SCARVTFI  REGIANO 

PER  FARE  RAGIONE.  ET  CONCORDANZA  D  ORo' 
E  I>ARGENTO;  CHE  SERVTRA  IN  VNIVERSALE; 
TANTO  PER  PROVEDERE  A  GU  INFINITI  ABVSI 
EEL  TOSARE,  ET  GVASTARE  MONETE;  Q^MsJTO 
PER  RZGOLARE  OGNI  SORTE  DI  PAGAMENTl 
ET  RIDVRRE  ANOO  TVTTO  IL  MONDO 
ADVNA   SOLA   MONETA. 


Fig.    184.    Title  page  ok  scaruffi 


372  RARA   ARITHMETICA 

Editions.    There  was  no  other  edition. 

A  verbose  commentary  on  certain  parts  of  Scaruffi's  work.  See 
p.  370. 

JOANNES  THOMAS  FREIGIUS. 

Ed.  pr.  1582.  Basel,  1582. 

A  Swiss  educator  of  the  sixteenth  century. 

Title.  '  loan.  Thomae  //  Freigii  I.  V.  D.//  Paedagogvs.//  Hoc 
est,  libellvs//ostendens,  qva  ratio-//ne  prima  artivm  ini-//tia 
pueris  quam  facilli-//me  tradi  pof-Z/fint.//  Basileae,//  per  Sebas- 
tianvm//Henricipetri.'    (P.  i.) 

Colophon.  '  Basileae,// per  Sebastianvm  Hen-//ricpetri,  anno 
salvtis//noftr£e  inftauratae  CI3.  13.  XXCn.//Menfe  Septem- 
bri.'    (P.  383-) 

Description.  8°,  10. i  X  15.8  cm.,  the  text  being  6.8  X  12.4 
cm.  17  pp.  unnumb.  +  i  blank  +  366  numb.  =  384  pp.,  30-3 1  11. 
Basel,  1582. 

Editions.    There  was  no  other  edition. 

This  is  a  general  summary  of  the  subject-matter  of  education,  pub- 
lished, after  the  author's  death,  by  his  two  sons  John  Thomas  and  John 
Oswald  (Joannes  Osualdus)  Freigius.  The  section  devoted  to  arith- 
metic begins  on  p.  144  and  ends  on  p.  156,  and  p.  145  is  reproduced 
in  Fig.  185.  Only  the  fundamental  operations  with  integers,  fractions, 
and  compound  numbers  are  given,  save  for  nine  lines  on  the  '  Aurea 
Regula  .  .  .  uulgo  uocatur  regula  Detri.' 

MATTHEW  HOSTUS.     Ed.  pr.  1582.  Antwerp,  1582. 

A  German  educator  of  the  sixteenth  century. 

Title.    See  Fig.  186. 

ColopJion.  '  Matthaeus  Hoftus  Francofordiae  ad  Qderam  haec 
obferuata  congerebat  &  edebat  elegantioris  literaturae  //  ftudiofis 
gratificaturus.  Anno  Chrifto  nato  CI3.  13.  LXXXI.'  (On  map 
at  end.) 

Description.  8°,  10  X  14.6  cm.,  the  text  being  ^.6  X  13.3  cm. 
61  pp.  numb.  +  3  blank  =  64  pp.,  20-32  11.  Antwerp,  1582 
(colophon  1 581). 

Editions.    There  was  no  other  edition. 


PRINTED    BOOKS  373 


DE      AHITHMETTC  A.  H^ 

Quid' eft  AiithiiK'tica  ? 

T^fl  ars  bene  nmiera.idiSubicdi^m  Igitur  Arithmicie 
tfmmtm. 

Q2pt  funt  confidcrandain  nuiuero? 
Vuo:  }^ota(ioonumerailo. 

Quxnam  eft  Notatio  J 
UmminahacofcnbendicTnotandidtccmnotafunt^ 
,j_, .^.5. 6.7.8.9.0.  CircKfMi per /erMfjj!/i^rM//cdf,Hd- 
Ut  tmtn  ad  alien  notai  amplijiccidm  fro  uarijigrudibui. 
Htie.ioo.iooo.ioooo. 

Qusnam  fuerunt  notx  Roma* 
noriim? 
I.        I. 

X.      to. 

C.      xoo. 
^.  D.   lo.    ?oo-      Qiimgentd. 
CXD-     00.    C  I  D.    iooo.        XImx.  M'lUe. 

L.  103.    5000.      Quf'ti^Mcmj/fj*/. 

CMd.  v^  .  CCiDD. 10000.  uJzioe.Deccmrrnllidt 
k  ,  iD3D-$°°°0'   Q}tlnc^uagmtamilU(i. 
J,  .  CCCIooD-     to6ooo.C«n<fmmiI/id* 
i-^dO.  <ooooo.Q}<wgen<4wJf^rf. 

CCCCIDD3D.  4.  CCCClDODD.xOo6ooo.Dmej 

nomam  nmen  non  j;>rogredmtur  ultra  decks  cemen<t 
milliaillancuplurdfigmficareuoUmt,dufUcamr.otAv.ut, 

CO  .     <*■•  1000. 
CIO.  ClO-   CIO.    30««' 
B'  CIO,  10.  ijoo.    CO  •  D. 

k  loo 

Fig.   1 85 .    From  the  Fccdagogvs  of  fkeigius 


374 


RARA   ARITHMETICA 


D  E 

NVMERATIONE 

EMEI^DATA, 
VETERIBVS     LATINIS 

tx'GRitClS    VSITATA, 

<5Matth<co  HoHo  auoiore. 


Antverp  i/e, 
Ex  ofHcina  Chrifloplum  Plancini. 

M.    D.    L  X  X  X  1   I.       0 

Fig.  1 86.    Title  page  of  hostus 


PRINTED    BOOKS  375 

This  is  a  semi-historical  treatise  on  the  various  numeral  systems  found 
in  Renaissance  literature.  It  includes  the  Arabic  system,  and  the  (Jreek, 
Latin,  and  Hebrew  ('  gens  ludaica,'  as  Hostus  sjjeaks  of  it)  systems, 
and  a  chapter  '  I)e  notis  Numerorum  Aftronomicis  (luibufdam  vfitalis,' 
a  set  of  mediaeval  astrological  numerals  also  fully  dcscritjcd  by 
Noviomagus. 

MAFFEO  POVEIANO.     Ed.  pr.  1582.  Bergamo,  1582. 

A  Veronese  arithmetician  of  the  sixteenth  century. 

Title.    See  Fig.  187. 

Description.  4°,  15.1  X  16.5  cm.,  the  text  being  10.3  x  15.6 
cm.    92  ff.  (83  numb.),  22-26  11.    Bergamo,  1582. 

Editions.    There  was  no  other  edition. 

An  ordinary  treatment  of  the  fundamental  operations,  with  a  few 
applications  to  mercantile  affairs.  The  book  had  not  enough  merit  to 
warrant  a  second  edition.  The  second  part  of  the  work  treats  of  ele- 
mentary mensuration.  The  book  is  little  known,  and,  like  many  others 
in  this  list,  is  not  mentioned  by  De  Morgan. 

Other  works  0/1582.  Gemma,  p.  200,  1540;  Moya,  p.  310,  1562  ; 
Sacrobosco,  p.  32,  1488  ;  Stevin,  p.  386,  1585  ;  Clement,  '  Summa  del 
arte  arithmetica,  de  Fr.  de  Sant  Clement,'  Barcelona,  4°  ;  Ognibene  de 
Castellano,  '  II  lineamento  i)ertinente  aU'  intendere  facilmente  quello, 
che  Euclide  &  altri  Eccellentiss.  Mathematici  ha  trattato  oscuramente,' 
Vincenza,  8°  (contains  some  theory  of  numbers  ;  see  also  p.  306,  1561) ; 
Mellema,  '  Arithmetique  composte  de  plusieurs  inventions  et  problemes 
nouveaux,'  Antwerp,  2  vol.,  1582  and  1586. 

CHRISTOPHER  CLAVIUS.     Ed.  pr.  1583.       Rome,  1583. 

Christoph  Ki.AiT.  Born  at  Bamberg  in  1537;  died  at  Rome,  February  6, 
161 2.  He  was  a  Jesuit  priest,  and  taught  mathematics  in  the  Jesuit  college 
at  Rome.    He  wrote  a  number  of  treatises  on  mathematics. 

Title.    See  Fig.  188. 

Description.  8°,  10.5  X  16.6  cm.,  the  text  being  8  X  13.3  cm. 
219  pp.  numb.  +  13  unnumb.  =  232  pp.,  38  11.    Rome,  1583. 

Editions.  Rome,  1583,  8°  (here  described) ;  Cologne,  1584; 
Rome,  1585,  8°  (p.  378);  Cologne,  1592.  There  were  also  edi- 
tions after  1600  (see  p.  378  for  the  1602  edition).  The  Italian 
translation  of  1586  is  mentioned  on  p.  378.  The  collected  works 
of  Clavius  in  five  volumes  appeared  at  Basel  in  161 2,  fol. 


2,1^ 


RARA   ARITHMETICA 


FATTOKE 

LIBRO  D'ARITFIMETICA. 

ET  GEOMETRIA  PRATTICAXE. 

DI  M.  MAFFEO  POVEIANO 

VERONESE. 

Opera  noua ,  &  vtiliflima-> 

Vdk  pl^brm  y&  generdli  pTAttkhe ,  che  ufar  ft  poffano  i 
necejpirie  ad  ogrimo . 


•      Con  Hcenza  de'  Supcrlori. 


IN     BERGAMO     LANNO     DI     N.     SIG. 
M     D     L  X.X  X  I  I. 

Per  Comln  Ventura ,  Stampatorein  cila  Citta. 
Fk;.   187.    Title  pa(;e  of  poveiano 


PRINTED    HCKJKS 


in 


CHRISTOPHORI 

C  L  A  V  I  I 
BAMBERGENSIS 

E    SOCIETATE 

I  E  S  V 

BPirOME    ^ RltHME- 
ticx  Pra6itca, 


PER.MISSV    SVPEarORVM. 

^OM*AE   Ex Typogr aphid D om'mk'i B.tfa,  1 5 g j . 


Fig.   188.   Title 


PAGE    OF    CLAVIUS 


378  RARA   ARITHMETICA 

Clavius  was  an  excellent  teacher  of  mathematics,  and  his  textbooks 
were  models  of  good  arrangement.  This  work  is  an  attempt  at  a  prac- 
tical arithmetic.  It  is  conservative  in  treatment,  the  applications  being 
confined,  as  was  the  custom,  largely  to  the  rule  of  three.  It  was  too 
scholarly  to  be  popular  in  schools  under  the  mercantile  influence,  but 
it  was  influential  in  the  classical  schools. 

CHRISTOPHER  CLAVIUS.     Ed.  pr.  1583.       Rome,  1585. 

See  p.  375. 

Title.  '  Christophori // Clavii // Bambergensis //e  Societate 
//  lesv  //  Epitome  Arithmeticae  //  Practicae  nunc  denuo  ab  ipfo 
auctore//recognita.//  (Woodcut  with  I.  H.  S.  in  center.)  Permis- 
sv  Svperiorvm // Romae  Ex  Typographia  Dominici  Bafae.  1585.' 
(P.  I.) 

Description.  8°,  10.3  X  16  cm.,  the  text  being  8x  13  cm. 
337  PP-  (321  mmib.),  31  11.    Rome,  1585. 

See  above. 

CHRISTOPHER  CLAVIUS.     Ed.  pr.  1583.       Rome,  1586. 

See  p.  375. 

Title.  '  Aritmetica  //  Prattica  //  composta  dal  Molto  //Reuer. 
Padre  Chriftoforo  Clauio  //  Bambergenfe  della  Com-//pagnia  di 
I  E  S  V.//Et  tradotta  da  Latino  in  Italiano  dal  Signor// Lorenzo 
Caftellano  Patritio// Romano.// Con  Licentia  dei  Svperiori.// 
(Woodcut  with  I  H  S  in  center.)  In  Roma,//  Nella  Stamperia  di 
Domenico  Bafa.//M.  D.  LXXXVL'    (P.  i.) 

Description.  8°,  10.3  X  i  5.6  cm.,  the  text  being  8.2  x  13.3  cm. 
302  pp.  (275  numb.),  39  11.     Rome,  1586. 

Editio7is.    See  p.  375. 

This  is  merely  an  Italian  translation  of  the  1583  edition  (see  p.  375). 
Clavius  was  unable,  however,  to  popularize  the  book  in  the  mercantile 
schools  of  Italy,  although  several  editions  appeared  after  1600. 

CHRISTOPHER  CLAVIUS.     Ed.  pr.  1583.       Rome,  1602. 

See  p.  375. 

Title.  'Arithmetica//Prattica//composta  dal  Molto //Reuer. 
Padre  Chriftoforo  Clauio  //  Bambergenfe  della  Com-//pagnia  di 


PRINTED    BOOKS  379 

lESV.//  Et  tradotta  da  Latino  in  Italiano  dal  Si^^mor  Lorenzo// 
Caftellano  Patritio  Romano.//  Reuifta  dal  medemo  Padre  Clauio 
//con  alcune  aggiunte.//Con  Licentia  de  i  Svperiori.//In  Roma, 
//(woodcut  with  LH.S.  in  center.)  Per  li  Meredi  di  Nicolo 
Mutij//M.  DC.  IL'    (P.  I.) 

Colophon.    '  In  Roma,//  Per  li  Ileredi  di  Nicolo  Mulij  M  D  C 

II.'     (P.  312.) 

Description.  4°,  10.4  X  i  5.7  cm.,  the  text  being  8.3x1  3.3  cm. 
312  pp.  {281  numb.),  39  11.    Rome,  1602. 

Editions.    This  is  the  second  Italian  edition  (p.  375). 

See  p.  378. 

NICOLAUS  REYMERS.     Ed.  pr.  1583.  Leipzig,  1583. 

A   German  surveyor,  bom  at   'Henstede  in    Dietmar.schen.'    The   '  Be- 
schluss'  is  dated  '  zu  Hattftede  in  Diethmarchen,'  September  14,  1583. 

Title.  'Geodjesia//Ranzoviana.//  LandtRechnen///vnd  Feld- 
meffen/  fampt  meff en  aller-//hand  gr6ff e.  Alles  auff  eine  leichte/ 
behende/  //  vnd  vormals  vnbekandte  newe  art/  kunft-//lich/ 
grundlich  vnd  deutlich  //  befchrieben/  //  Zu  Ehren  //  Dem 
Edlen/  Beftrengen  //  vnd  Ehrnuehften  Herrn/  Heinrichen  // 
Rantzouen/  Herrn  Johans  feligen  Sohne/  der  //  Kon.  Mayft. 
zu  Dennemarcken/  etc.  In  den  //  Furftenthumben  Schlefewick/ 
Holftein/vnd  Diethmar-//fchen/Stadthaltern/ Rhat  vnd  Ambt- 
man  auff  //  Segeberge/  Erbgefeffen  zum  //  Breitenberge/  etc.// 
Durch//Nicolaum  Reymers/von  Henftede///in  Dietmarfchen. 
//Cvm  Privilegio.'    (F.  i,  r.) 

Colophon.  '  Gedruckt  zu  Leipzig  bey  //  Georg  Defner/  //  Im 
Iahr//M.  D.  LXXXIII.'     (F.  44,  v.) 

Description.  4°,  15.1  X  18.6  cm.,  the  text  being  9.7  X  14.5  cm. 
44  ff.  unnumb.,  25  11.     Leipzig,  1583. 

Editions.    There  was  no  other  edition. 

Although  nominally  a  book  on  surveying,  this  work  may  properly  rank 
as  an  arithmetic,  the  '  Erfte  Buch '  being  entirely  devoted  to  that  sub- 
ject. Of  this  book  the  first  chapter  is  entitled  '  von  zahlen  '  ;  the  second, 
'von  brfichen  '  ;  the  third,  'von  fummieren ' ;  the  fourth,  'von  viel- 
feltigen,'  multiplication  thus  directly  following  addition  ;  the  fifth.  '  von 


38o  RARA   ARITHMETICA 

abziehen  '  ;  the  sixth,  '  von  Theilen  '  (division)  ;  the  seventh,  *  von  den 
Wurtzel ' ;  the  eighth,  '  von  der  gevierten  Wurtzel.'  The  rest  of  the 
work  is  devoted  to  mensuration,  and  in  particular  to  surveying.  The 
chief  interest  in  the  first  book  is  in  the  use  made  of  the  compound 
numbers  then  needed  in  surveying.  Like  most  such  manuals,  it  shows 
no  insight  into  educational  problems,  and  the  treatment  is  very  unsatis- 
factory. 

Other  works  of  1583.  Baker,  p.  327,  1568  ;  Caesar,  p.  370,  1582  ; 
Gemma,  p.  200,  1540;  Petri,  p.  325,  1567;  Reisch,  p.  82,  1503; 
Anonymous,  '  Rechenbiichlein  auf  Erfurtifche  weifs,'  Erfurt,  8°  ;  Johnn 
Weber,  '  Ein  new  Kunstlich  Rechenbuch  auff  den  linien  und  ziffern,' 
Leipzig,  4°  (see  also  p.  338,  1570). 

PETRUS  BUNGUS.     Ed.  pr.  1583-84.      Bergamo,  1584-85. 

Bom  at  Bergamo  ;  died  September  24,  1 601,  at  Bergamo.     He  was  a  canon 
of  the  cathedral  in  that  city.    He  wrote  only  on  the  mystery  of  numbers. 

Title.  See  Fig.  189,  which  gives  the  title  page  of  the  first 
part.  Bound  with  this  is  the  second  part  with  the  title  :  '  Mys- 
ticae // nvmerorvm // significationis // pars  altera,// lo.  Petro 
Bongo  Canonico  Bergomate //avctore,// In  qua  de  Numeris  in 
Sacris  libris  potiffimum  repertis,  e.x  Theo-//logorum  ma.xime  fen- 
tentia,  &  probatorum  aliorum  cuiufuis  //  facultatis  Scriptorum, 
ita  exacte,  dilucide,  &  accurate  differi-//tur,  vt  ferme  nil  addi,  aut. 
detrahi  poffe  videatur://Opus  varia  fane,  et  multiiuga  adeo  re- 
fertum  doctrina,  vt  non  Theo-//logis  folum;  fed  etiam  Philofo- 
phis,  Mathematicis,  atque  alijs  ftuV/diofis  omnibus,  tam  vtile, 
quam  iucundum  fit  futurum.//De  Svperiorvm  licentia.// Bergo- 
mi  CID  ID  XXCIV.//Typis  Commini  Venturse,  eiufdem  Vrbis 
typographi.' 

Description.  Fob,  20.6  X  30.4  cm.,  the  text  being  17  x  28.9 
cm.  276  pp.  (245  numb.)  in  the  first  part ;  198  pp.  (177  numb.) 
in  the  second  part;  36  11.  Bergamo,  first  part  1585,  second 
part  1584. 

Editions.  The  first  edition  appeared  in  1583-84,  '  Bergomi, 
typis  Comini  Venturae.'  The  second  part  of  this  work,  as  the 
title  page  shows,  is  therefore  of  the  first  edition.  The  first  edi- 
tion of  the  first  part  was  evidently  exhausted  before  that  of  the 


M  Y  S  T  I  C  AE 

NVMERORVM 

SIGNIFICATIONIS 

LIBER 
IN  DVAS  DIVISVS  TARTES, 

R.    D.    PETRO     BONGO 

CANONICO    BERGOMATE 
A  V  C  T  O  R  E: 

Opus  maximarum  rerum  ,  &  plurimarum  do(ftrIna^,  fua- 
uirato,copia^,  &uarictate  refcrtuiiu, 

Theologis  ,Thilo[6phu  y  Mathematicis  y  atcfue  aliji  fiudiojis 
omnlbtiSj  tarn  vttlitatemy  quam  iumnditaternj 

DE  SVTERIORVM  LICENTIA, 


BERGOMI  CD  D  X  X  C  V. 

Typis  Comini  Vcncurae ,  ^Socij. 

Img.   1 89.    Title  pal;e  of  thk   1585   bungus 


382  RARA   ARITHMETICA 

second  part  was  sold,  and  hence  the  first  part  of  the  present 
work  is  the  second  edition. 

There  were  also  editions  as  follows:  Venice,  1585,  8°;  Ber- 
gamo, 1590;  ib.,  1 591,  4°  (which  De  Morgan  incorrectly  calls 
the  second  edition);  ib.,  1614  (p.  384).     It  appeared  under  the 


Romanorum 

CtD      CD      CD 

CO  ^      %. 
Ill 

IIIM 
CO  «' 

00       IDD 

^000. 
^000. 

fOOC. 

6000. 
7099. 

tooe. 
161 4    EDHION    OF 

V 

v' 

IDD 
IDD 
ICC 
VCD 

VM 

IDD  CO 
IDD  00 

VII  CO 

IDD    C.3  CD 

CID  ^'^  «'33 

Fig.  190.    From  the 

BUNGUS 

title  'Numerorum  mysteria,'  Bergamo,  1599,  4°  (p.  384).  I  have 
an  edition  under  this  title,  published  at  Paris  in  161 8  (161 7 
in  the  colophon),  4°.  For  the  'Praecipuae  numerorum  notae  et 
earum  valor,'  Parma,  1689,  see  p.  384. 

This  is  a  mass  of  erudition,  prolix  and  unscientific,  relating  to  the 
mystery  of  numbers.    It  was  written  by  a  priest  for  the  use  of  preachers, 


PRINTED    HOOKS 


383 


and  it  indudes  all  of  the  allusions  to  siu  li  nialtcrs  as  the  mystic  three 
that  Bungiis  could  find  in  ancient  literature.  He  takes  uj)  the  various 
numbers  from  one  to  ten  in  the  same  way,  together  with  a  few  of  the 
more  interesting  larger  numbers.  For  students  interested  in  popular 
number  mysticism  the  book  still  remains  the  classic  in  its  way.  It  is 
also  of  much  value  in  showing  the  nature  of  the  Roman  numerals  in  use 
in  the  sixteenth  century.    (See  Figs.  190,  191.) 


Fi( 


Numcratio. 

CD ''''' 

00  cciw 

pooo. 

CC[OD 

CCIDO 

X 

CCICC 

DMC 

0M3 

IMl 

10000. 

CCDD  ^'^ 

CCIOD  00 

CID 

CID  CD 

IIOOO. 

CCOD  ^'' 

CCIOD  ** 
CCIDD  ^  ^ 

12000. 

13009. 

CCOD  C.3 

CCI3D    00 

ID3 
133 

1 .4.000. 

CCDD  ^00 

191.    From  ' 

FHE     1 

15000.. 

614    EDITION 

Other  works  of  1^84.  Buckley,  p.  252,  1550;  Cassiodorus,  p.  211, 
1540;  Clavius,  p.  375,  1583;  Delfino,  p.  275,  1556;  Kobel,  p.  102, 
15 14;  Ramus,  p.  263,  1555  ;  Anton  Schulze,  '  Arithmetica  oder  Rech- 
enbiichlein,'  ( '  Arithmetica  oder  Rechenbuch  neben  einer  dienlichen 
Anleitung  zum  Buchhalten'?)  s.  1.,  with  subsequent  editions  at  Liegnitz, 
1600,  4°,  and  Frankfort,  1600,  4°. 


384  RARA   ARITHMETICA 

PETRUS  BUNGUS.     Ed.  pr.  1583-84.  Bergamo,  1599. 

See  p.  380. 

Title.  '  Petri  Bongi  //  Bergomatis  //  Numerorum  myfteria.// 
Opvs  maximarvm  rervm // Doctrina,  et  copia  refertvm,// In 
quo  mirus  in  primis,  idemq  perpetuus  Arithmeticae  Pythagoricae 
cum  //  Diuinai  Paginse  Nvmeris  confenfis,  multiplici  ratione  pro- 
batur.//Poftrema  hac  editione  ab  Auctore  ipfo  copiofo  Indice,  & 
ingenti//Appendice  avctvm.//Cum  Superiorum  approbatione.// 
Bergomi,  Typis  Comini  Venturse,  eiufdem  vrbis  Typographi.// 
(Xl3  XCIX.'     (P.  I.) 

Description.  4°,  17.5  X  24.3  cm.,  the  text  being  12.6  x  18.8 
cm.     770  pp.  (676  numb.),  32  11.     Bergamo,  1599. 

See  p.  382.  Although  the  title  is  slightly  changed,  this  is  the  work 
already  described. 

PETRUS  BUNGUS.     Ed.  pr.  1583-84.  Bergamo,  1614. 

See  p.  380. 

Title.  '  Petri  Bvngi  //  Bergomatis  //  Nvmerorvm  myfteria  // 
Ex  abditis  plurimarum  difciplinaru  fontibus  haufta://Opvs  maxi- 
marvm rervm //Doctrina,  &  copia  refertum :  In  quo  mirus  in  primis, 
idemq;  perpe-//tuusArithmeticaePythagoricecumDiuinaePaginae 
Nu-//meris  confenfus,  multiplici  ratione  probatur.//Poftrema  hac 
editione  ab  Auctore  ipfo  copiofo  Indice,  &  ingenti  //  Appendice 
avctvm.// Illuftriffmio  viro,  Virtutum  omnium,  ac  difciplinarum 
//genere  ornatifQmo  //Ranvtio  Gambarae  //  Comiti  Virolae.  &  c. 
// Bergomi,  Typis  Comini  Venturse.  1614.'    (P.  i.) 

Description.  4°,  17.5  X  23.4  cm.,  the  text  being  12.6  X  18.8 
cm.     970  pp.  (753  numb.),  32  11.    Bergamo,  1614.  _ 

See  p.  382. 

PETRUS   BUNGUS.     Ed.  pr.  1583-84.  Parma,  1689. 

See  p.  386. 
Title.    'Praecipuae  //  Numerorum //Notae,//  Et  earum  valor,// 
Secundum  //  Petrum  //  Bungum.//Parmae,  Ex  Typographia  Du- 
cali.//CL3.  I3C.  LXXXIX.'    (P.  i.) 


PRINTED    BOOKS 


385 


Dcscripiioii.  4°,  i  i.i  X  17  7  cm.,  the  text  hcin<;  7.9  X  12.6cm. 
16  pp.  (12  numb.),  13-19  11.     rarma,  1689. 

Editions.  See  p.  380.  This  is  simply  an  extract  from  the  later 
editions  of  the  '  Numerorum  Mysteria.' 

The  treatment  of  Roman  numerals  by  Bungus  is  the  most  elaborate 
and  interesting  to  be  found  in  any  of  the  works  of  Renaissance  writers. 


MONTE  REGAL  PIEDMONTOIS. 

Ed.  pr.  1585.  Lyons,  1585. 

The  title  states  that  he  was  professor  of  mathematics  in  the  University  of 
Paris;  evidently  in  the  second  half  of  the  sixteenth  century. 

Title.    See  Fig.  192. 

Description.  32°,  5.3x9.8cm., 
printed  on  vellum  in  double  col- 
umns, each  1.8  X  9.5  cm.  144  pp. 
(ioonumb.),27-2811.  Lyons,  1585. 

Editions.  There  was  no  other 
edition.  The  privilege  is  dated 
August  6,  1 581,  but  the  work 
does  not  seem  to  have  been  pub- 
lished until  1585.  The  book  is  ex- 
ceedingly rare,  and  is  unknown  to 
most  bibliographers.  The  author 
speaks  of  having  published  part 
of  the  tables  in  Venice  in  1575. 

This  is  a  collection  of  tables,  largely 
for  multiplication,  beautifully  printed 
on  split  vellum.  These  tables  give  the 
products  of  numbers  to  100  times 
1000.  The  last  ten  pages  contain  a 
table  of  arrangement  of  soldiers  in 
order  of  battle  :  '  Le  moyen  et  ordre 
ciui  fe  doit  tenir  pour  mettre  en  or- 
donnance  les  batailles  de  dix  soldats 
iufques  au  nobre  de  quarante  mille.' 
This  is  dedicated  '  A  pvissant  et  illustre 
Seigneur  le  Baron  de  Mont-clar.' 


jlNVENTlONl 

NOVVELLE    ET 

admirable, pour  faire  route 
forte  dc  coptc  ,  tant  de  mar- 
chandife ,  commc  dc  chager 
monnoyes,p«ids,mefures,dc 
dinerfe  mauicrcpoucvendre 
I  &  acheter,laqiieUc  feruiia  en 
I  tour  le  monde  auec  grande 
facilite,fans  gecos  ne  plume. 

1    £f  a.  Ufn  le  moyen  de  meiire  vn 
I       exercU  en  batatlte  ,  de  cent 


iHjfjues  a  quatunte  mille. 


iLe  lout  noiiucllcmfut  compofe  8c 
j  mis  en  Utmieie  par  le  Moote  Re- 
•     gal  Picdmontois  ,  Profelfeut   de 

Mathemaaque  en  I'Vaiuerfitide 

Vm'h. 


A     L  Y  O  N> 
!  lile  vend  enrueMcrciere  i 
I      Tcnfcgne  de  la  SpKcie. 

J Auec  pr'iuY.c^c da  Uny, 

Fig.  192.  Title  page  of  monte 
regal  piedmontois 


386  RARA   ARITHMETICA 

SIMON   STEVIN.     Ed.  pr.  1585.  Leyden,  1585. 

Born  at  Bruges  in  1548;  died  at  the  Hague  in  1620.    He  was  a  merchant, 
a  soldier,  and  an  officer  in  the  civil  service. 

Title.    See  Fig.  193. 

Description.  8°,  10. 5  X  16  cm.,  the  text  being  j .6  x  12.7  cm. 
3 1  pp.  unnumb.  +  i  blank  +  642  numb.  +  (203  numb.  +  1 2  unnumb. 
+  I  blank  in  '  La  Pratiqve  d'Arithmetiqve  ')  =  890  pp.,  29  11. 
Leyden,  1585. 

Editions.  Stevin's  first  work  was  an  interest  table,  Antwerp, 
1582.  His  arithmetic  first  appeared  in  Flemish,  at  Leyden,  1585, 
and  was  reprinted  in  that  language  at  Gouda  in  1626  and  in  1630. 
The  French  translation,  which  is  here  described,  appeared  at 
Leyden  in  1585  and  again  in  1586.  The  first  edition  under  the 
editorship  of  Girard  appeared  at  Leyden  in  1625.  For  the  edi- 
tion of  1634  see  below. 

This  work  consists  of  three  distinct  parts  :  (i)  '  L'Arithmetiqve,'  in 
two  books,  the  first  treating  of  powers  and  roots,  and  particularly  of 
surds,  and  the  second  of  operations  on  numerical  and  algebraic  expres- 
sions and  of  the  solution  of  equations  ;  (2)  '  Les  qvatre  premiers  Livres 
d'Algebre  de  Diophante  d'Alexandrie,'  translated  by  Stevin,  apparently 
from  Xylander's  text  ;  (3)  *  La  Pratiqve  d'Arithmetiqve,'  an  attempt  at 
a  practical  textbook,  but  too  scholarly  for  its  purposes.  The  Pratiqve 
contains  'La  Regie  d'Interest  avec  ses  tables,'  the  1582  work  above 
mentioned,  '  La  Disme.  Enfeignant  facilement  expedier  par  nom- 
bres  entiers  fans  rompuz,  tous  comptes  fe  rencontrans  aux  affaires  des 
Hommes.  Premierement  defcripte  en  Flameng,  &  maintenant  conuertie 
en  Frangois,'  and  a  '  Traicte  des  incommensvrables  Grandevrs.'  The 
interest  centers  in  '  La  Disme,'  in  which  decimal  fractions  are  for  the 
first  time  treated  in  any  elaborate  way  (see  Fig.  194). 

SIMON  STEVIN. 

Ed.  pr.  of  the  arithmetic,  1585.  Leyden,  1634. 

See  above. 

Title.  '  Les //CEuvres//Mathematiques//de// Simon  Stevin, 
//Augmentees// Par  Albert  Girard.'    (P.  i.) 

Page  3  reads  :  Les  //  CEuvres  //  Mathematiques  //de  Simon 
Stevin  de  Bruges.// Ou  font  inferees  les  // Memoires  Mathema- 
tiqves,//  Efquelles  s'eft  exerc6  le  Tres-haut  &  Tres-illuftre  Prince 


PRiNri<:i)  I'.ooKs 


387 


I'ARITHMETIQJVE 

DE   SIMON   STEVIN 
DE     BRVGES: 

Conrcnant  Ics  computations  des  nombres 

Anthmctiques  ou  vulgaires : 
Auft  ['Ahebr:,  auecles equations de cine  qudntitez^. 
Enlemblclesquarrc  premiers  liures  d'Algebrc 
dc  Diophante  d'Alcxandric,maintenant  pre- 
miercmcnt  craduidls  en  Fran9ois. 

Encore  vn  linrepamculier  de  la  Pratique  d'Aritbmetique, 
(omen^::t  cntre  amres,Les  Tables dlniere^, La  Dtfm; 
Et  vn  tr.iicle  des  hicommenfm.ibles  (irandcms  : 
Aucc  i Explnatiou  dit  Dixiefme  Liure  d'Eudide. 


A     L  E  Y  D  E, 

Dc  I'lmpiimcriedeChiiftophlc  Pkitin* 
do.  Id.   lxxxv. 

Fig.   193.    Title  paop:  of  thk    1585  stkvin 


388 


KARA   ARITHMETICA 

SECONDE     PARTIE     DE 
JLA    DISME    DE    L'OPE, 

It  A  T  I  O  N, 

PRpPOSlTION   I,    DE 

L'A  D  D  I  T  I  O  N. 

EStant  domez.  nombfes  de  D'lfme a ajonJier :  Trouver  Uut 
fomme : 

Expfkatioh  du  donni.  II  y  a  trois  ordres  de  nombres  de, 
Difmc,  dcfquels  le  premier  17  i@8  ® 4 @7 (CU^  deux- 
iefmc  57  @  8  ®  7  (i)5®,le  troiiipfme  875  @7  (y)S@t®, 

Explication  durequis.  Ilnousfaut  /-\r>r^r\ 

On  mettra  les  nombres  donnez  a  7  8  4  7 

ea  ordre  coramc  ci  joignant ,  les  ^  ^       Z   ^ 

aiouftant  felon  la  vulgaire  manierc  ^757^1 
d'aiouftcr  nombres  entiers-jCncefte  p  4  *  5  <^  4 
forte: 

Donne  (bmme  (par  Ic  i  problems  de  rArithmeti- 
que)  941304,  qui  font  fee  que  dcmooftrent  les  (ignes 
deflus  ics  nombres)  941  @  3  ®  o  Q)  4®.  Ic  di,  que^ 
les  mefmes  font  la  fomme  reqiiifc.  Dmen^ration.  Les" 
27  (o)  8  ®  4  @  7  0  donnez,  font  (par  la  36  definition^ 
^7  1%,  jh^  TZ->  enfcmble  zj  ^^^,  ^J^'^^^^f^ 
raifon  les  37  (0)6  (i)  7  @  5  ©  salient  37  rs^,  &  les 
87  5  @  7  ®  8  @4  ®  feront  87s  rs'^»  lefqucls  trois 
nombres,comme  27  At?,  37  -Hh,^  ^{5  irh^/^J^^ 
cnfemble  Cpar  le  io<=  piobleme  de  1  Arith.}  941  -fSVo* 
nuis  autant  rauc  aufli  la  fomme  94»  ®  5 0  o  0  4  (L\ 

cefF 
Fig.    194.   From  the   1634  edition  of  stevin 


PRINTl'I)    BOOKS  389 

Maurice //dc  Nassau,  Prince  d'Aurenge,  Gouverneur  des  Pro 
vinces  des  //  Pais-bas  unis,  General  par  Mer  &  i)ar  Terre,  &c.// 
Le  tout  reveu,  corrige,  &  augmente  //  Par  Albert  Girard  Samie- 
lois,  Mathematicien.// A  Leyde  //  Chez  Bonaventure  &  Abraham 
Elfevier,  Imprimeurs  ordinaires  //de  rUniverlUe,  Anno  cLt  Io  c 
xxxiv.' 

Description.  Fol.,  21.7  x  34  cm.,  printed  in  double  ccjlumns, 
each  being  8.1  x  28  cm.  910  pp.  (232  on  arithmetic,  10  unnumb.), 
63  11.     Leyden,  1634. 

Editions.    See  p.  386. 

Other  works  of  1585.  Benedetti,  p.  364,  1580;  Bungus,  p.  382, 
1583  ;  Caesar,  p.  370,  1582  ;  Clavius,  p.  378,  1583  ;  Uigges,  p.  340, 
1572;  Gemma,  p.  200,  1540;  Lonicerus,  p.  253,  1551;  Lossius, 
p.  289,  1557  ;  Psellus,  p.  168,  1532  ;  Riese,  p.  139,  1522  ;  Savonne, 
p.  314,  1563  ;  lo.  Frans.  Fulconis, '  Cisterna  Fulconica,  libro  d'abaco  in 
lingua  provenzale,'  s.  1.  (Lyons?),  8°;  Thilman  Ofenlach,  '  Rechen- 
biichlein  mit  der  Ziffer  und  auf  den  Linien  mit  Zahlpfennigen,'  Basel, 
8°  ;  Johann  Schreckenberger,  '  Rechenbuchlein  auff  den  Linien  und  der 
Federn,'  Strasburg,  8° ;  Samuel  Eisenmenger  (pseud.  Siderocrates), 
*  Cyclopaedia  Paracelsica  Christiana,'  s.  1.,  with  a  section  on  arithmetic. 

Works  of  1586.  Clavius,  p.  378,  1583;  Gray,  p.  353,  i577  ; 
Mellema,  p.  375,  1582  ;  Ramus,  p.  331,  1569;  Riese,  p.  139,  1522; 
Schonerus,  p.  331  (Ramus,  1569);  Stevin,  p.  386,  1585;  Stifel, 
p.  226,  1544  ;  Tagliente,  p.  115,  15 15  ;  Anonymous  (Sterner  mentions 
a  Rechenbiichlein  without  title  page,  Magdeburg)  ;  (ieorg  Hciflein, 
'  Rechenbiichlein  mit  der  Ziffer  und  mit  den  Zahlpfennigen  auf  der 
Linie,'  Strasburg,  8°  ;  Paulus  Alexandrinus,  '  Rudimenta,'  Greek  and 
Latin,  Wittenberg,  4°  (with  some  notes  on  Jewish  arithmetic). 

JOHANNES  PADOVANIUS.     Ed.  pr.  1587.    Verona,  1587. 

Giovanni  Padovani.    A  Veronese  mathematician  of  the  second  half  of 
the  sixteenth  century. 

Title.    Sec  Fig.  195. 

Description.  4°,  14.4x20.4  cm.,  the  text  being  9.6  X  15.7 
cm.  7  pp.  unnumb.  +  73  numb.  =  80  pp.,  29-32  11.  Verona, 
1587. 

Editions.  Padovani  published  a  work  in  Venice  in  1565  con- 
sisting of  six  parts,  the  fourth  f)f  which  was  '  De  Arithmetica.' 


390 


RARA   ARITHMETICA 


lOANNIS  PADOVANII 

VERONENSIS, 

DE  ARITHMETICA  OPVS,IN  QVO  NON 

folum  omnis  generis  numerandi  ars  am  Latino 

rermone,quam  Grxco  pcrdifcitur:  ucrum 

ctiam  qiiicquid  ad  quafcunque  ra- 

tiocinationes  pertinet ,  facili 

doftrina  apcritur. 

Omnibus,  &in  primisreipuhlkaliteraria  ftudiofts  perneceffariunu 


V  E  R  O  N  AE, 


ExTypogmphia  Sebaftiani  a  Donnis.  15S7. 
Delicentia  Siiperiorum. 

Fio.   195.    Title  page  of  padovanius 


PRINTr:i)    HOOKS  391 

That  work  is  probably. the  same  as  this.    This  seems  to  be  the 
first  and  only  separate  edition,  and  is  very  rare. 

This  is  in  no  sense  a  practical  treatise,  but  it  discusses  the  subject  of 
arithmetic  in  a  learned  way,  at  first  rather  on  the  Greek  plan.  It  then 
considers  the  Arabic  arithmetic,  taking  up  the  four  processes,  roots,  pro- 
portion, partnership,  and  two  or  three  other  applications.  'l"he  style 
in  which  it  is  written  is  such  that  it  would  hardly  have  appealed  to  the 
practical  merchant  even  if  it  had  not  been  in  Latin.  Yor  Padovani's 
other  works  see  Riccardi,  I,  i,  251. 

CASPAR  THIERFELDER. 

Ed.  pr.  1587.  Nurnber£(,  1587. 

A  German  Rechenmeister  of  Steyer,  born  c.  1525. 

T/t/r.    See  Fig.  196. 

ColopJion.  '  Gedruckt  zu  Number^/  //  durch  Leonhardt  // 
Heufzler.'     (P.  369.) 

Description.  8°,  9.4  X  1 5.2  cm.,  the  text  being  7  X  13.3  cm.  30 
pp.  imnumb.  +  369  numb.  =  399  pp.,  25-26  11.    Nurnberg,  1587. 

Editions.  There  was  no  other  edition.  Thierfelder  also  pub- 
lished a  book  with  Ulman  at  Freiburg  in  1 564,  8°,  entitled  '  Neues 
Kunft-Rechenbuch  auf  der  Linie  und  Feder  ;  dergleichen  weder 
in  lateinifchen  noch  deutfcher  Sprache  ausgegangen.' 

This  is  a  commercial  arithmetic,  based  on  Rudolff  and  other  Ger- 
man writers,  and  with  no  particular  individuality.  It  contains  eighteen 
chapters,  the  last  one  being  upon  mathematical  recreations,  *  Von  der 
Schimpff  Rechnung/vnd  Erfindung  derfelben  Regeln,'  a  subject  that 
began  to  come  into  prominence  about  this  time. 

Other  works  of  ijSy.  Finaeus,  p.  160,  1530-32  ;  Riese,  p.  139, 
1522;  Michael  Gempelius,  '  Arithmetik,'  s.  1.,  8°;  Aurelio  Marinati, 
'  Delia  prima  parte  della  somma  di  tutte  le  scienzc,'  Rome,  4°  (with  a 
brief  treatment  of  arithmetic)  ;  Conr.  Poeppingius,  *  Neues  Rechen- 
biichlein  auf  I  Jnien  und  P'edern,'  Braunschweig,  8°  (four  editions  after 
1600). 

Works  0/1588.  Gemma,  p.  208,  1540;  Helmreich,  p.  306,  1561  ; 
Rudolff,  p.  152,  1526;  John  Mellis,  'A  brief  instruction  and  maner 
how  to  keepe  bookes  of  Accompts,'  London,  8°  (containing  a  short 
chapter  on  arithmetic)  ;  Heinrich  Striibe,  '  Arithmetica  oder  new  kiinft- 
hches  Rechenbuchlein,'  Zurich,  8°,  with  an  edition  ib.,  1599,  12°. 


392 


RARA   ARITHMETICA 


ej^>^-^'^ 


gfuff  bm  2mm  m^  "^iffmi 

mit1iott^tt)hnb  ^t^mU^fclvanf 

Cur*  i2^%^^ 
©e&rucf  1 5u  7?i5mkr3/b«rc$ 

M.  D,  LXXX.VII, 

Fig.   196.    Title  page  of  thierfelder 

BERNAERT  STOCKMANS. 

Ed.  pr.  1589.  Dordrecht,  1609 

A  French  schoolmaster  at  Dordrecht  (Dort),  m  the  second  half  of  the  six- 
teenth century. 

7>V/r.    '  Een  corte  ende  een-//vuldige  Inftructie/om  lichtelij- 
cke  //  en  by  hemfelve/  fonder  eenige  meefter  oft  onder\vij-//fer 


PRINTia)    HOOKS  393 

te  Iceren  cijffercn.  Secp  nut  en  protijlclijckcn  //alio  nienfchen/ 
die  in  de  Confte  van  Arith-//metica  heel  ftccht/oncrvaren // 
ende  eenvuldich  zijn.//  Geftelt  endc  by  ccn  vcrgadert,  door 
Bernaert // Stockmans  lanfz.  F"ran9oyfche  Schoolmeefter //inde 
vermaerde  Coop-ftadt  Dordrecht.// Hier  zijn  ooc  bygevoecht  de 
Differentien  van  de  co-//i'en-Mate  der  voornaefte  ftede  in  IIol- 
lant/  t'Sticht/  //  Zeelant/  Brabant/  Vlaenderen/  Gelderlant/ 
En-//gelant/  Vrancrijck/  eii  Ooftlant/  tot  dienft  //  van  alien 
Graen-coopers.// Van  nieus  overfien  ende  verbetert  door  C.  P. 
Boeye.//  Item/  noch  van  nieus  bygevoecht  een  clare  onder-// 
wyfinge  om  de  tafelen  van  Intereft  te  leeren  maken/  //alles  tot 
dienft  vanden  onervarenen.//  Sapiente  1 1.  verf.  22.//  Maer  ghy 
hebbet  alles  in  Mate/  Tellen///  ende  Ghewichte  gheordineert.// 
Tot  Dordrecht  //  By  my  Pieter  Verhagen/  woonende  inde  Drue 
// kerije/teghen  over  de  Wijnbrugge.  1609.'     (F.  i,  r.) 

Colophon.  'Tot  Dordrecht.// By  Adrien  lanfz  Bot.//Anno 
1609.'     (F.  214,  v.) 

Description.  8°,  9.  i  X  14  cm.,  the  text  being  7.4  x  11.9  cm. 
4  ff.  blank  +  211  unnumb.  =  215  ff.,  31-33  11.    Dordrecht,  1609. 

Editions.  This  book  was  first  printed  in  1589,  and  the  dedi- 
catory epistle  is  dated  '  den  20.  defer  Maent  Julij  /  1 589.'  There 
was  no  other  edition  in  the  sixteenth  century. 

This  is  one  of  the  noteworthy  arithmetics  of  Holland,  and  it  went 
through  a  number  of  editions  after  i6or  (see  p.  394).  It  is  a  commer- 
cial work,  and  like  Vander  Schuere's  book  it  gives  an  e.xcellent  view  of 
the  mercantile  life  of  Holland  in  this  period. 

Other  jvorks  of  i^Sg.  Jacob,  p.  298,  1560;  Moya,  p.  310,  1562; 
Fabio  Paolini  (Paulinas),  '  Hebdomades,  sive  septem  de  septenario 
libri,'  Venice,  4°,  containing  a  little  arithmetic,  and  ib.,  1598  (?). 

Works  of  1590.  Bungus,  p.  382,  1583;  Digges,  p.  340,  1572; 
Jacob,  p.  298,  1560;  Lapazzaia,  p.  324,  1566;  Psellu.s,  p.  170,  1532; 
Recorde,  p.  214,  c.  1542;  Heizo  Buscher  (Boscherus),  'Arithmetics 
libri  duo,'  Helmstadt,  sm.  8°,  with  editions,  ib.,  1591,  8°;  Hamburg, 
1592,  8°;  ib.,  1597,  8°;  Frankfort,  1600,  8°,  and  later;  Franciscus 
Brasser,  an  arithmetic,  Liibeck,  with  later  editions  ;  Cyprian  Lucar,  '  A 
treatise  named  Lucarsolace,'  London,  4°;  Bartolom^  Solorzano,  '  Librode 
caxa  V  Manual  de  cuentas  de  Mercaderes,'  Madrid. 


^     OF    THE 

UNIVERSITY 


394  RARA   ARITHMETICA 

BERNAERT  STOCKMANS.     Ed.  pr.  1589.      Gouda,  1644. 

See  p.  392. 

Title.  '  Aritmetica,//  Door  Bernardus  Stockma  //  eertyts 
Franfoyfche  fchool-//meefter  inde  vermaerde  coopftadt  //  Dor- 
derecht  nu  van  nieu*  curieus  //  gecorigert  ende  verbetert  noch  is 
//  hier  by  gevoecht  een  tafelken  cm  te  //  rabatteeren  op  fulcken 
tyt  of  te  intereft  //  men  begeert  door  //  Abel.  W.  Waesenaer  // 
Rekenmeefter  tot  Vtrecht  //  (Portrait  of  author.)  Gedruckt  tot 
Vtrecht  by //Efdras  Willemfsen  Snellaert.//boeckvercooper // 
Anno.  1637.'     (P.  I.) 

Colophon.  '  Ter  Govde,//  Gedruckt  by  Pieter  Rammazeyn, 
Boeck-// drucker  in't  vergulde  A  B  C.//Door  Efdras  Willemfz 
Snellaert,  Boeck-//vercooper  tot  Vtrecht  by  de  waert-poort  1644.' 

(P.  421.) 

Description.  8°,  9.3  X  13.9  cm.,  the  text  being  7.5  X  12.4  cm. 
429  pp.  (410  numb.),  32-33  11.    Gouda,  1644. 

Editions.  See  p.  393.  The  engraved  title  page  of  the  1637 
edition  has  been  used  with  this  edition,  which,  as  the  colophon 
shows,  was  printed  in  1644. 

See  p.  393. 

FRANCESCO  PAGANI.     Ed.  pr.  1591.  Ferrara,  1591. 

An  Italian  arithmetician  of  the  sixteenth  century,  born  at  Bagnacavallo. 

Title.    See  Fig.  197. 

Description.    4°,  14.2  x  17.9  cm.,  the  text  being  10.  i  X  15.9 
cm.    210  pp.  (200  numb.),  38  11.  Ferrara,  1591. 
Editions.    There  was  no  other  edition. 

This  rare  and  almost  unknown  work  is  based  upon  the  Borghi  model 
(see  p.  16)  which  served  for  so  many  of  the  best  Italian  writers.  It 
was  written,  as  the  dedicatory  epistle  states,  at  Bagnacavallo,  and  is  one 
of  the  few  books  on  mathematics  published  in  the  sixteenth  century  at 
Ferrara.  It  has  no  merit  saVe  as  its  applied  problems  give  a  view  of 
the  business  life  of  the  time.  In  its  numerical  work  it  is  reactionary, 
making,  for  example,  a  strong  plea  for  the  galley  as  opposed  to  the 
'  a  danda '  or  modern  method  of  division. 


PRINTED    HOOKS 


395 


ARITHMETICA 

PRATTICA   VTILISSIMA, 

ARTIFICIOSAMENTE  ORDINATA 

Da  M.Francesco  PAGANidaBagnacauallo, 

J^Ua  quale  ft  cont'tene  il'vero^  ^facile 
modo  di  conteggiarc^ . 

Conmoiti  Quefiti  importanti,&i>eccnanj 

a  Ragionieri,a  MercantijcS:  ad  ogni 

per{ona,in  tutti  i  Paefi. 

^L  mOLTO  JLLVSTRE  SIGNOR 

il^  F  F^  ELE     K^  STO  2^/,  &c. 


(T^^^ 


(/  M  c-Tcn*./ 


ApprefTo  Vittorio  Baldini 

Con  liccnza  dc'  Supcriori.  M.  D.  xci. 


'ic.  197.    Title  page  of  pag 


396  RARA   ARITHMETICA 

RENE  BUDEL,  et  al.     Ed.  pr.  1591.  Cologne,  1591. 

For  biographies,  see  below. 

Title.    See  Fig.  198. 

Description.  4°,  18.2  x  23.6  cm.,  printed  in  double  columns, 
each  being  6.4  X  18.6  cm.  38  pp.  unnumb.  +  798  numb.  = 
836  pp.  (PP.  271-350  are  not  in  the  volume,  and  if  they  were 
ever  bound  in  any  copies  they  must  have  constituted  a  section  by 
themselves.)    46-49  11.    Cologne,  159 1. 

Editions.    There  was  no  other  edition. 

This  rather  massive  treatise  on  the  history  of  monetary  measures  con- 
sists of  two  books  by  Budel  (Budelius),  director  of  the  Bavarian  mint, 
and  several  appended  chapters  by  the  following  writers  :  Albertus  Brunus 
(1461-1541),  counselor  to  Louis  of  France,  and  ambassador;  Johannes 
Aquila,  friend  of  the  astrologer  Stoffler  who  died  in  1 5  3 1  ;  Bilibaldus 
Pirkheymer  (1470-1530),  a  celebrated  humanist;  Martinus  Garatus 
Laudensis,  who  writes  a  chapter  *  De  monetis  ' ;  Franciscus  Curtius, 
and  Joannes  Regnaudus  of  Avignon,  who  write  on  the  same  topic ; 
Caroius  MolinKus  (1500-1566)  ;  Didacus  Covarrubias  (1512-1577), 
bishop  of  Ciudad  Rodrigo ;  Henricus  Mameranus,  a  Belgian  printer ; 
Henricus  Hornmannus ;  Franciscus  de  Aretio  (1418-1483),  a  cele- 
brated lawyer  of  Arezzo ;  Nicolas  Everardus  (1473-1532),  a  celebrated 
Dutch  lawyer,  of  Middelburg;  Jacobus  Menochius  (i 531-1607),  an 
Italian  lawyer,  and  various  others.  It  is  a  monumental  work,  and  is 
helpful  in  the  investigation  of  the  history  of  monetary  tables. 

Other  works  of  ijgi.  Baker,  p.  327,  1568  ;  Bungus,  p.  382,  1583  ; 
Buscher,  p.  393,  1590;  Gemma,  p.  200,  1540;  Mariani,  p.  181,  1535  ; 
Petri,  p.  325,  1567;  Psellus,  p.  168,  1532;  Ramus,  p.  331,  1569; 
Vincent  de  Beauvais,  p.  10,  1473;  Jordanus  Bruno,  '  De  Monade 
numero  et  Figura  liber,'  Frankfort,  8° ;  Johann  Kaudler,  '  Arithmetica 
Oder  Rechnung  auf  der  Linien  und  mit  Ziffern,'  Regensburg,  8° ;  Hans 
Jacob  Mewrer,  '  Bericht  von  dem  Rechnen  mit  den  Zahlpfennigen  oder 
auf  der  Linien,'  Zurich,  8°. 

THOMAS   HYLLES.     Ed.  pr.  1592.  London,  1600. 

An  English  mathematician  of  the  latter  part  of  the  sixteenth  century. 

Title.    See  Fig.  199. 

Description.  4°,  14.2  X  19.6  cm.,  the  text  being  10. 8  X  16.9 
cm.  I  5  ff .  unnumb.  +  270  numb.  =  285  ff.,  with  2  charts  at  the 
end;  40  11.     London,  1600. 


PRINTED    BOOKS  397 

MONETIS, 

ET  RE  NVMA- 

RIA,    LIBRI    DVO: 

QVORVM    PRIMVS   ARTEM    CVDENDAE    MO 

NETAE:    SECVNDVS    VEHO    QVAESTIONVM    MO- 
NETARIAKVM     DECISIONBS     CoNTINET. 

HIS  ACCESSERVNT   TRACTATVS   VARII   ATQVE 

VTILES,    NECNON     CONSlLIA,SINGVLARF.S(iyH     ADDI- 

tionc  tam  vcternm, quain Neotericorum  Authorum.qui 

deMonctis,c3rundcmqiievalore,!iga,pondere, 

poteftate,mutationc,variatione,fallitate, 

aciimilibusfcripferunt. 

Quorum  oimiium  Catalo^umpa^hu  duodecinu  indicat. 

AVTHORE     ET     COLLECTORE     CLARISS. 

Vjro   Renero    Bvdelio    Rvremvndano,  IC.    Necnon    Reve- 

readirs.atqucIlkiftrifs.Principisac  D.Domini  Erncfti  Elcdloris 

Colonien  BauariaeDuc.&c.  MonetarLim,tani  Rhc 

nciilium,  quam  V vcftphaticanim  Archie, 

pifcopaliiim  Prsfcdo. 

CVM     SVMMARIIS     ET     InDFCE     COPIOSO. 


COLONIAE     AGRII'PINAE» 

APVD     lOANNEM     GYMNICVAT, 

SVB     MONOCEROTE. 
ANNO    M.    D.    L  X  X  X  X  I. 

CVM    PRIVILEG.   CAES.  MAIEST.  AD   SEXENNIVM. 

Fig.  198.   Title  page  of  budel  et  al. 


398 


RARA   ARITHMETICA 


The  Arte  of  vulgar  arith- 

meticke ,  both  in  Integers  and  Fradions, 
dcuidedinto  two  "Boo/^esnvhereofthefirU  is  called 

UomodidacltisNumerorum,2.u(X  the  fecond  PoriusFroportiomnj.  vvitlx 
cciteineDemonftrationSji educed  into  fo  plaine  and  perfedMc- 
thod,as  the  l\e  hath  not  huhmo  beene  fuhhfhed  in  Engliih  •  Wherevvto 
j  saddcd  a  third  BookCjCmituIcd  t-^1/^ -^f'^iT '^<"'«W.  Com- 
prehending aJl  the  moll  rcceflo-rie  and  profiiableKuks 
vfed  in  the Udde  tfMcrcbaidifc, 

In  all  which  three  Bookes :  the RuIeSjPrcceptSjandMaximSjare not 

cnclj  com^ofedinmeeter for  the  better  ret  a'mtng  oj them  in  mcmor'te^ 

but  alfo  the  operanons,  cKaniples.denionlirationSj  and  qucftions, 

axe  in  Tnofheafie  wife  ocpoHnded  and  ex flaned,itt  the  forme 
ofa  Diahcucfoi  the  Readers  more  dcert  v(-dcrltar  ding. 

i^  hcivhdgeple/ijantfor  GentUmev,  comniendahlefor  Captuna 

and  Soldiers,  profitable  for  Merchants,  and  e^cnerally 

mcejfatie  for  allejlatex  and  degrees. 

Newly  colIe(5led,digefted,  and  in  fomc  part  deuifed  by 

a  '^el'^tlUr  to  the  MathematicaU, 

Ecdcfiafticus,    cap.  iff. 

Learning  vntof  odes  is  as  fetters  on  their  feeteandManiclcs  vpcn 
thcirnghthand:  buttothc  wifcitisalcwcllofgoldcjandhke 
a  Bracelet  vpon  his  right  arme. 

Boctiuslibr.  i.  Arith.  cap.  t» 
Omnia^juacutique  afrlmma  nAturaconflritEiafHtit^NuMmruw  videmw 
radmefoiwaeafHoccmTifMtprinctpalewarjfmocondmuexiimflar, 


Imprinted  at  London  by  Cal;riclSimfatt,dVic]!iit^g 
ioflceiclanc,   xtfoo. 

Fig.  199.   Title  page  of  the   1600  hylles 


PRINTED    HOOKS  399 

Editions.    London,  1^92;   ib.,  1600,4°  (here  dcscribctl). 

This  rather  ponderous  work  of  570  pages  is  written  in  the  form  of  a 
dialogue,  following  the  popular  textbooks  of  Recorde.  Hylles  intro- 
duces his  rules  and  definitions  in  verse,  presumably  with  the  idea  that 
they  can  be  more  easily  memorized  in  this  form.  An  illustration  of  this 
feature  is  seen  in  the  following  description  of  the  first  case  of  '  Barters 
or  Truc([ues,'  a  chapter  now  obsolete,  but  one  of  whic  h  we  have  a 
reminder  in  the  words  '  barter '  and  '  truck  '  : 

Of  Barters  or  trucciucs,  there  are  diuers  kindes, 
Whereof  the  firft,  is  when  the  Trucquers  take. 
But  ware  for  ware,  by  agreement  of  mindes, 

No  partie  grating,  greater  gaines  to  make, 
Thequalitie  of  which  exchange  of  wares. 

The  compound  rule  afcending  fole  declares.'  (F.  255.) 

The  author  seems  to  take  up  every  rule  known  to  the  English  arith- 
meticians of  the  time,  and  his  book  is  a  good  source  of  information 
concerning  British  commerce  of  the  period.  It  was  not  a  popular  work, 
probably  because  it  elaborated  its  rules  too  much  to  be  usable. 

JOANNES  ANTONIUS  MAGINUS. 

Ed.  pr.  1592.  Venice,  1592. 

Giovanni  Antonio  Magini.  Born  at  Padua,  June  13,  1555;  died  at 
Bologna,  February  11,  1617.  He  was  professor  of  astronomy  and  mathe- 
matics at  Bologna,  and  wrote  numerous  works  on  these  subjects. 

Title.  '  lo.//  Antonii  //  Magini  Patavini  //  Mathematicarvm 
in  almo  //  Bononienfi  Gymnafio  prof eff oris.//  Dc  Planis  Triangv- 
lis  //  Liber  Vnicus.//  Eitifdem  //  de  Dimetiendi  ratione  //per 
Quadrantem,  &  Geometricum  Quadratum,//  Libri  Qvinqve.// 
Opus  valdevtile  Geometris,  Aftronomis,  Geographis,  Mechanicis, 
Ar-//chitectis,  Militibus,  Agrorum  menforibus,  &  denique  //om- 
nibus Mathematicarum  profefforibus.// Cvm  Privilegio.// Vene- 
tiis,  apvd  Robertvm  Meiettvm.// M.  D.  XCIL'     (F.  i,  r.) 

Colophon.  '  Bononiae,//  apud  lannem  Baptiftam  Ciottum.// 
Typis  Victorij  Benacij,// Anno  Domini,  M.  D.  XCn.//Superi- 
orum  permiffu.'     (F.  114,  v.,  and  132,  v.) 

Description.  4°,  1 6  X  2 1 .6  cm.,  the  text  being  1 1 .8  X  18.2  cm. 
4  ff.  unnumb.  +  110  numb,  (in  the  first  book)  +  124  numb,  -f-  4 


400  RARA   ARITHMETICA 

unnumb.  (in  the  second  book)  =  242  ff.,  34-38  11-    Venice  (but 
printed  in  Bologna,  as  the  colophon  states),  1592. 
Editions.    There  was  no  other  edition. 

Following  some  other  bibliographers,  I  have  included  this  treatise  on 
plane  triangles,  because  of  the  '  Expositio,  ac  vsvs  tabvlae  tetragonicae, 
feu  Quadratorum  numerorum  cum  fuis  radicibus  iuxta  fequentes  octo 
Canones  '  (f .  5 ,  r.)  which  it  contains.  The  '  Tabvla  nvmerorvm  quadra- 
torum '  begins  on  f.  41  and  extends  to  f.  64,  inclusive.  It  is  the  most 
extensive  table  of  squares  and  roots  that  had  appeared  up  to  1592. 

THOMAS  MASTERSON. 

Ed.  pr.  1592-95.  London,  1592-95. 

An  English  mathematician  of  the  latter  part  of  the  sixteenth  century. 

Title.    See  Fig.  200. 

Description.  4°,  12.8  X  18.  i  cm.,  the  text  being  10.  i  x  15.6 
cm.  255  pp.  (230  numb.),  34-35  11.  London,  1592-95.  Book 
II  (148  pp.)  bears  the  date  1592  ;  Book  III  (78  pp.)  bears  the 
date  1595. 

Editions.  There  was  no  other  edition.  There  was,  however, 
an  addition  to  the  '  First  Booke  '  published  at  London  in  1594. 

The  author  says  in  his  dedicatory  epistle,  dated  '  London  this 
20  Auguft  Anno.  1592,'  that  he  has  '  vnder  taken  to  write  and  publifhe 
fixe  bookes  of  the  Art  of  Arithmeticke  :  with  this  order  and  methode, 
that  the  firft,  third,  and  fift  bookes,  fhalbe  as  a  fummarie  and  ground, 
teaching,  the  true  ingenious,  inuentions,  and  the  perfect  figuratiue  and 
caractericall  operations  of  the  fame  Art  .  .  .  Then  the  feconde,  fourth, 
and  fixt  bookes,  fhalbe  of  Arithmeticall  queftions  and  demaundes,  with 
the  application  of  the  definitions,  common  fenteces  and  inftructions 
of  the  firft,  third,  and  fift  bookes.'  The  plan  was  not  carried  out  beyond 
the  publication  of  the  three  books  and  the  supplement  mentioned  above. 

Book  I  is  on  the  fundamental  operations  with  integers  and  fractions, 
and  makes  no  advance  in  the  ordinary  textbooks  of  the  period. 
Book  II  is  a  collection  of  practical  problems  representing  the  mer- 
cantile activities  of  London  at  the  close  of  the  sixteenth  century 
(see  Fig.  201).  Book  III  would  now  be  classed  as  part  of  algebra  (see 
Fig.  202),  since  it  refers  chiefly  to  irrational  numbers. 

Othei-  works  of  isg2.  Buscher,  p.  393,  1590  ;  Capella,  p.  68,  1499  ; 
Clavius,   p.  375,   1583;    Fischer  (Piscator),   p.   247,    1549;  Gemma, 


PRINTED    HOOKS 


401 


THOMAS 

MASTERSON    HIS 

FIRST      BOOKE 
OF    ARITHMETICKE. 

Shewing  ihc  ingenious  inucntions ,  and  figuratiuc  opera- 
tions ,  by  which  to  caiculaie  the  tnie  folution  or  anfwcres 
of  Arithmetical!  quelHoos:  after  a  more  perfect,  plainc, 
briefc,  well  ordered  Anthmecicall  way  ,  then  any 
other  heretofore  publifhed :  vcric 
ncccflarie  for  all  men. 

Nothing  without  labour. 
\^ll things  with  reafon. 


Imprinted  at  London  by  Richard  Ficld,dvvelling  in 

the  Blacke  nriers  ncare  Ludgatc.  ^ 

o'c.3  4-- 

FiG.  200.  Title  page  of  masterson 


i5Pi. 


402  RARA   ARITHMETICA 

p.  200,  1540;  Nonius,  p.  315,  1564;  Psellus,  p.  168,  1532;  Ramus, 
p.  331,  1569;  Riese,  p.  139,  1522;  Tartaglia,  p.  279,  1556;  Rutilio 
Cosentino  Benicansa,  '  Corona  di  tutte  le  scientie  de  Abaco,'  Naples, 

lit  THOMAS    MASTERSON    HIS 


5  • 

.  200 

'  •      3 

Facit 

IZO 

200  . 

•  4  . 

.    800 

100 

300  . 

«  4  . 

.  1200 

2JJL. 

150  . 

.  4  . 

.   600 

80   .  . 

.  2600 

'~  X20 

Facit 

3900. 

400 . 

*  S  •  '_ 

2000 

15)00 
27  li 

Facit  B  tooke  out  i  2  8  ^  pound. 
J38  Two  maichants  made  a  companic,  A  put  in  300  pound  for  2 
monethes,  and  then  putteth  yet  in  1 00  pound ,  and  6  monctlies 
after  that  taketh  out  200  poud,and  with  the  reft  remaincth  vn- 
till  the  yeares  end.B  put  in  1 00  pound  for  one  moneth,and  then 
putteth  yet  in  700  pound ,  and  6  monethes  after  that  taketh  out 
acertaine  fnmme  of  money  ,  and  with  the  reft  remaineth  vntill 
the  yeares  end.and  then  finde  to  hauc  gained  togcther4oo  poud, 
whereofBmufthaue  80  pound  more  then  A  ,  thequeftion  is 
how  much  money  B  tooke  out  of  the  companie,  without  recke- 
ring  intcreft  vpon  intercft. 


3  CO   . 

.  2  , 

.  .     600 

400 

I  00 

80 

400  . 

.  6  . 

.  2400 

320 

2  00 

I  60 

200  . 

•4. 

,     800 

160     , 

.   3T00  , 

.  .  240 

_}J9_ 

5700 

I  00  .    . 

.1  . 

.      1 00 

7^00 

8oo«  • 

6j_ 

.4800   FacU  B  tooke  out 

^40  pound. 

4900 
800 
160 

If 

Fig 

.  20 

I .    From 

MASTERSON 

8°  ;  Julius  Caesar  of  Padua,  '  Gevviffe  Erinnerung  einer  allgemeinen 
arithmetifchen  Practic,'  Cologne,  16°  (see  also  p.  368,  1582)  ;  Johann 
Krafft,  '  Kin  neues  vnd  wohlgegriindtes  Rechenbuch,'  Ulm. 


PRINTIlD    1KK)KS 


^  THOMAS     MASTERSON    HIS 

hauin''  here  their  number  written  oUcr  them. 
I.  2.    ^    4.     ?•    ^'      7-      8.       9.       lo.    II.     12. 

The firftcaraacr  written  thus  z^,  doeth  fignific  anynum- 
bcr  before  which  it  is  written,  to  be  the  firft  number  giuen, 
taken,  or  imagincd-,and  is  called  radix,  or  rootcfor  that  all  the 
other  caraftcrshaue  their  original!  or  of-(f>ring  of  ir.  The  fc- 
cond  written  thus^,  is  called  ^^^-^  or  fquarc,  and  doth  fignifie 
any  number  before  which  it  is  written,  to  be  the  produd  of  the 
firft  multiplication  of  the  roote  by  it  fclfe :  that  is  of  the  roote 
two  times  taken  and  multiplied.  The  third  written  thus  c^, is 
called  cn^e,  and  doth  fignific  thenumber  following  the  fame 
to  be  the  produft  of  the  fecond  multiplication  of  the  roote.three 
times  taked  and  multiplied,  that  is  of  the  ^  multiphed  by  the 
z^.  The  fourth  is  called  z^nje:zen/e^  and  dothfignifie  the  num. 
bcr  following  the  fame,to  be  the  produft  of  the  third  multipli- 
cation,that  is  the  produft  of  thc^  fourc  times  taken  and  mul- 
tiplied. The  fift  is  called /«r/S/;ii/<f,  and  doth  fignific  the  number 
following  the  fame,  to  be  the  produft  of  the  fourth  raultiplica- 
tion.  The  fixt  is  called  Kenfecnbe ,  and  doth  fignific  thenumber 
following  the  fame,to  be  the  produft  of  the  fiift  multiplication. 
The  feuenth  is  called  bfurfolide ,.  or  {econd  furfolidc ,  and  doeth 
fignifie  the  number  following  the  fame,to  be  the  produft  of  the 
fixt  multiplication.  The  eight  is  called  KjenfcK^nfez^enfe^  6c  doth 
fignific  the  number  following  the  fame,to  be  the  produft  of  the 
feuenth  multiplication.  The  ninth  is  called  cubecube ,  and  doth 
fignifie,  the  number  following  the  fame,to  be  theprodu£fc  of  the 
eight  multiplication.  Xiie  tcntli  is  called  i^enfefurfoUde,  &  doth 
fignifie  the  number  following  the  fame ,  to  be  theprodud  &c. 
The  eleuenth,  is  called  cfurfolide.ox  third  furfblide.Thc  twelfth 
is  called  Kjenfez^enjecube.  The  thirteenth  is  called  d/urfolide^  or 
fourth  furfolide.  You  may  proceede  further  at  your  pleafure,  if 
you  marktthat  the  fift  carafter  is  the  firft  furfolidc,  the  feuenrh 
the  fecond  furfolidc,  and  thenexcvncompound number  fol- 
lowing,thcnextfurfolide,5c  fo  infinitely  cucr  tlie  next  vxicom- 

pound 
Fig.  202.    Fkom  masterson,  showing  algebraic  sv.mbols 


403 


404  KARA   ARITHMETICA 

SIGISMUNDUS  SUEVUS.     Ed.  pr.  1593.        Breslau,  1593. 

A  German  priest,  living  at  Breslau.    Born  c.  1550. 

Title.    See  Fig.  203. 

Colophon.  '  Gedruckt  zu  Brefslaw/  durch  //  Georgium  Baw- 
mann/J.  Jn  Mitvorle-//gung  Andreae  Wolcken.  Jm  lahre  :// 
M.  D.  XCIIL'    {P.  523) 

Description.  4°,  14.8  X  19.5  cm.,  the  text  being  11.2  X  15.8 
cm.     524  pp.  (455  numb.),  28-31  11.    Breslau,  1593. 

Editions.    There  was  no  other  edition. 

This  work  was  probably  intended  to  be  a  practical  arithmetic,  but 
the  author's  theological  interests  unfitted  him  for  the  task  of  writing 
such  a  book.  Although  the  fundamental  operations  and  the  common 
rules  of  the  day  are  treated  in  somewhat  the  usual  way,  the  problems 
are  largely  biblical,  or  mystical.  The  size  of  Goliath's  armor  and  Gematria 
used  to  foretell  the  famine  of  Poland  represent  the  applications. 

ALESSANDRO  ALAMAGNI. 

Ed.  pr.  1593.  Venice,  1593. 

A  Venetian  arithmetician  of  the  sixteenth  century. 

Title.    See  Fig.  204. 

Description.  12°,  8.7  X  15.4  cm.,  the  text  being6.3  X  13.2  cm. 
I  f.  unnumb.  +  96  numb.  =  97  ff.,  printed  in  double  columns, 
30  11.    Venice,  1593. 

Editions.    There  was  no  other  edition. 

The  word  '  tariffa '  in  these  early  books  means  a  table  used  by  mer- 
chants to  assist  in  their  computations  involving  the  measures  and  awk- 
ward monetary  systems  of  the  time.    See  Paxi,  p.  77  ;  Mariani,  p.  180. 

Other  works  of  i^gj.  Gemma,  p.  200,  1540;  Tartaglia,  p.  279, 
1556  ;  Horatio  Galasso,  '  Giochi  di  Carte,'  Venice,  8°,  with  an  edition 
at  Verona  in  1597,  12°,  and  a  French  translation  in  160^  (containing 
several  '  giochi  d'abbaco  '). 

MIGUEL  GERONIMO  SANTA  CRUZ. 

Ed.  pr.  1594  Madrid,  1643. 

A  Spanish  merchant  and  arithmetician  of  the  second  half  of  the  sixteenth 
century,  bom  at  Valencia.    He  lived  at  Seville. 

Title.  '  Libro  de  //  Arithmetica  //  especvlativa,  y  pra-//tica, 
intitvlado,  el  Dorado  //  Contador,  contiene  la  fineza  y  reglas  de 


PRINTKD    BOOKS 


405 


RITHMETICA 

HISTORICAL 
©Mrd&  a\W  Species  t?nt)  fflrnrmafe  9?rfldn/ 

im't  f(5ottctt  ^it)ct\(tttmm  ^ifictkn  mi  gjccmp.crtt  / 

auclj  inie  ^fbraircfjf r  /  ©ridjifc^r/ Dnb  tK^miTc^fr  9J?anep/@f« 

(»ic^  onD  eSjap/  Derm  in  ^filigcr  (Scftnfff  PnD  ^uffrn  ©fFc^ic^r^ 

QSilcfjfrn  gfDof^c  wirb/  5)fr  lictm  3ugtn&  iugutu  (ttltrsu 

3Iucfe  6fnm  Die  nid^t  rrcfenm  Wrnim/  Wf^rn  eir(fr 

am  lufiiB  f  nJ)  liV^lic^  Jii  Irfftt* 


am  Freyftadicnrem,      ^^^ 


Sigifmundum  Sueuum  irreyi 

pf^rtljcrc  ja  S.  Bernardin  tn  Dec  ncai|lt;ot. 
OnD  @croiC^(C-    Sapientai, 

CVM   GRATIA    ET    PRIVILEGIO. 

Fig.  20^.    Title  vxc.v.  of  suevus 


4o6 


RARA   ARITHMETICA 


T  A  R  I  F  F  A 

N  V  O  V  A> 

T>  E  L  L  A    V  ALVT  fi^ 

dull  Cccchini ,  ^j  Lire 

dieci ,  efaldt  vno^ 

Fino  a  ^  12  /  8.  Laqual  fcriC 

anco  per  oy;ni  (brtedi 

Mcrcantie,- 

FATTA     DA     M. 
Alcflandro  Alamagni. 


O^  ^ret*c4- 


IN    V  E  N  E  T  I  A, 


AppicfTo  Gio.  Ant.  Raaip.uecto.  15  pj. 
Fig.  204.  Title  page  of  alamagni 


PRINTED    BOOKS  407 

contar//()roy  plata,  y  los  Aneagcs  de // Flandcs.//Por  moder- 
no  y  com-//pcndiolo  cl'lil<).//Compvesto  por  Migvcl  Gero-//nim() 
de  Santa-Cruz,  natural  de  la  Ciudad,  y // Reino  dc  Valencia,  y 
vezino  de // Seuilla.// Al  L^o  Don  Pero  de  Baraiz,//Teniencc 
mayor  de  Corregidor  de  la  infigne  Villa  de // Madrid,  Corte  del 
Rei  nueftro  Senor.//Ano  1643 //Con  licencia.//  En  Madrid,  Por 
Francifco  Martinez. //A  coftade  luan  BautiftaTabano,  Mercader 
de  libros,  vendcfe  al  // lado  del  Colegio  de  Atocha.'    (F.  i,  r.) 

Description.  4°,  13.8  X  20  cm.,  the  text  being  9.9  X  16.8  cm. 
6  ff.  unnumb.  +  238  numb.  =  244  ff.,  32-34  !!■    Madrid,  1643. 

Editions.  I  know  of  no  sixteenth-century  edition,  but  the  orig- 
inal privilege  is  dated  May  8,  1594.  This  edition  of  1643  seems 
by  the  dedication  of  Tabano  to  have  been  published  after  the 
author's  death.    There  was  an  edition  as  late  as  1794. 

The  work  follows  the  general  plan  of  the  Spanish  and  Italian  mer- 
cantile  arithmetics  of  the  sixteenth  century.  It  is  not,  however,  a  very 
practical  book,  the  author  having  been  too  much  influenced  by  the 
theoretical  side  of  works  like  Tartaglia's. 

Other  works  of  I^g4.  Barlaamo,  p.  343-  i5  72  ;  Masterson,  p.  400, 
1592  ;  Recorde,  p.  217,  c.  1542  ;  Reisch,  p.  82,  1503  ;  Wenceslaus,  p. 
421,  1599  ;  Thomas  Blundevile,  'His  Exercises,  containing  sixe  Trea- 
tises,' London,  4°  (the  first  being  on  arithmetic  ;  there  was  an  edition  ib., 
1597,4°,  and  others  appeared  after  160:)  ;  Smiraldo  Borghetti,  'Opera 
d'abbaco,'  Venice,  8°  ;  Jerdnimo  Cortes,  'Arithmetica  practica,'  Valencia, 
Works  0/1595.  BelU,  p.  343,  1573;  Helmreich,  p.  306,  1561  ; 
Masterson,  p.  400,  1592  ;  Urstisius,  i).  361,  1579. 

JOANNES  BILSTENIUS.     Ed.  pr.  1596.  Basel,  1596. 

A  si.xteenth-centuvy  German  educator. 

Title.  Syntagma  //  Philippo-//rameum  //  Artium  Li-//bera- 
lium,  Methodo  brevi  ac  per-//fpicua  cominnatum //per // loan. 
Bilstenium  //  Marfbergianum.//  In  //  Gratiam  Tyronum  partim 
difficilibus  //  vocibus  Germanica  adiecta  //  est  explicatio.// 
Basilese,// Typis  Conr.  W^aldkirch.// CI3  ID  XCVI.'    (F.  i,  r.) 

Description.  8°,  10  X  16.  i  cm.,  the  text  being  6.8  X  12  cm. 
16  pp.  unnumb.  +  586  numb.  =  602  pp.,  26-28  11.     Basel,  i  596. 

Editions.    There  was  no  other  edition. 


4o8  RARA   ARITHMETICA 

This  work  consists  of  twenty  parts,  treating  respectively  of  the  lead- 
ing branches  of  knowledge  as  considered  in  the  sixteenth  century.  Of 
these  the  eleventh  is  '  I)e  Arithmetica,'  and  gives  in  43  pages  a  succinct 
account  of  algorism.  It  begins  with  the  catechism  form  :  '  Quid  est 
Arithmetica?  Arithmetica,  die  Rechenkilft/est  ars  bene  numerandi. 
Subjectum  Arithmeticae  eft  Numerus.'  There  is  nothing  progressive  in 
the  treatment,  division,  for  example,  being  performed  by  the  galley 
method  only,  and  the  applications  being  confined  largely  to  the  '  Aurea 
Regula'  (the  'Golden  Rule'  of  three). 

Other  wo7'ks  of  1596.  Helmreich,  p.  303,  1561  ;  Hood  (translator  ; 
see  Urstisius),  p.  361,  1579  ;  Petri,  p.  325,  1567  ;  Ramus,  p.  330, 
1569  ;  Recorde,  p.  219,  c.  1542  ;  Snellius  (see  Ramus),  p.  333,  1569  ; 
Tartaglia,  p.  278,  1556  ;  Urstisius,  p.  361,  1579  ;  Anonymous  (William 
Parley  (?),  translator),  '  The  Pathway  to  Knowledge,'  London,  4°  (con- 
taining 'Thirty  days  hath  September,'  see  p.  33)  ;  Anonymous,  '  Arith- 
metica,' Frankfort,  8°  ;  Sebastian  Brandt,  '  Plenaria  artis  Arithmeticae 
refolutio,'  Frankfort,  8° ;  C,  M.  Glysonius,  '  Arithmetica  practica,' 
Venice,  4°,  with  editions  as  late  as  1783  ;  Giacomo  Trevisano,  '  Me- 
moriale  di  abbaco,'  Venice,  8°  (title  page  1597,  colophon  1596)  ; 
Bernardo  Vila,  '  Reglas  breves  de  Arithmetica,'  Barcelona,  sm.  8° ; 
Antonio  Rodriguez,  '  Aritmetica  pratica  y  theoretica,'  Salamanca,  8°. 

ANONYMOUS.     Ed.  pr.  1597.  Luyck,  1597. 

Title.  '  Cleynen  //  Catholyken  //  Catechismvs.// . . .  Tot  Luyck 
by  Jan  Voes.// M.D.XCVII.// Met  priuilegie  van  fes  laren.' 
(F.  I,  r.) 

Colophon.    'Tot  Luyck  by  Ian  Voes,  1597.'     (F.  12,  v.) 

Description.  4°,  9  X  14  cm.,  the  text  being  6.2  x  11.8  cm. 
12  ff.  unnumb.,  16-21  11.     Luyck,  1597. 

This  is  one  of  many  similar  children's  books  in  Mr.  Plimpton's  library. 
It  is  introduced  here  merely  as  a  type  of  those  primers  that  taught  some 
work  in  number  in  connection  with  grammar  or  the  catechism.  In  this 
book  the  child  learns  the  Roman  and  Arabic  numerals  to  84.  There 
are  also  in  the  library  numerous  Latin  and  Greek  grammars  of  the  six- 
teenth century  in  which  numeration  is  taught,  but  these  have  been 
omitted  from  this  bibliography. 

Other  works  of  i^g"/.  Blundevile,  p.  407,  1594  ;  Bonocchio,  p.  346, 
1574  ;  Buscher,  p.  393,  1590  ;  Galasso,  p.  404,  1593  ;  Gemma,  p.  200, 
1540;  Trevisano,  above,  1596;  Francis  Meres,  'God's  Arithmetique,' 
London,  8°. 


PRINTED    HOOKS  409 

HENRY   DE  SUBERVILLE.     Ed.  pr.  1598.        Paris,  1598. 

A  French  scholar  of  the  close  of  the  sixteenth  century.  lie  describes 
himself  as  canon  of  the  cathedral  at  '  Xaintes,'  and  '  Aduocat  en  la  Cour  de 
Parlement  de  Bourdeaux,'  and  signs  his  name  with  the  birthplace  '  Hreton- 
Bearnois.'    The  dedicatory  epistle  is  dated  at  '  Kimpercorentin.' 

Title.  '  L' Henry-metre,// instrvment  royal,  ct // vnivcrscl, 
avec  sa  theoriqve,//  vsage,  et  pratiqve  demonstree  par  //  les  Pro- 
pofitions  Elementaircs  d'Euclide,  &  regies  familieres  // d'Arith- 
metique :  &  auffi  fans  Arithmetique :  Lequel  prend  toutes 
mefures  Geometriques,  8z  Aftronomiques,  qui  luy//font  circu- 
lairement  oppofees  tant  au  Ciel,  qu'en  la  Terre,//svr  vne  sevle 
station,  par  vn  //  feul  triangle  Orthogone,  fans  le  bouger  de  fa 
place,  ny  aller  //  mefurer  aucune  diftance  de  ftation,  ainfi  qu'on 
eft  con-/Arainct  de  faire  auec  les  autres  Inftruments  Geome- 
triques.//De  I'inuention // D'Henry  de  Suberuille  Breton,  Cha- 
noine  en  I'Eglife  Cathedrale  S.  Pierre  //de  Xaintes  :  &  Aduocat 
en  la  Cour  de  Parlement  de  Bourdeaux.//  Item,//  Vn  petit  traicte 
fur  la  Theorique,  &  Pratique  que  de  1' Extraction  des  racines 
quarrees,//pour  dreffer  les  Scadrons,  &  Bataillons  quarres.// 
Dedies  au  Roy.//  Diev  a  difpofe  toutes  chofes  en  Poids,  Nombre, 
&  //  Mesvre.  Sap.  11.  2 1 .//  A  Paris,//  Chez,  Adrien  Perier,  rue 
fainct  laques  en  la //boutique  de  Plantin  au  Compas.//  1598.// 
Auec  Priuilege  du  Roy.'    (F.  i,  r.) 

Description.  4°,  17  X  22.1  cm.,  the  text  being  1 1.8  x  17.8  cm. 
39  pp.  unnumb.  +  225  numb.  =  264  pp.,  39-42  11.     Paris,  1598. 

Editions.    There  was  no  other  edition. 

Although  this  is  primarily  a  treatise  on  mensuration  by  the  use  of  a 
trigonometric  instrument  somewhat  resembling  a  quadrans,  it  is  properly 
included  in  this  Hst  because  it  contains  several  chapters  on  fractions 
and  denominate  numbers.  These  are  of  no  special  merit,  and  are 
introduced  as  a  preliminary  to  the  calculations  involved  in  the  use  of  the 
'  Henry-metre '  which  the  author  invented.  The  book  is  an  interesting 
effort  to  perpetuate  an  inventor's  name  by  a  work  of  no  special  scholar- 
ship describing  an  instrument  of  no  particular  value.  There  is,  however, 
a  value  in  the  general  study  of  all  of  these  early  instruments.  Many  of 
them  are  easily  constructed,  the  quadrans  for  example,  and  their  more 
extensive  use  in  the  teaching  of  trigonometry  would  be  very  heli)ful. 


4IO 


RARA   ARITHMETICA 


JOHANN  FRIDOLIN   LAUTENSCHLAGER. 

Ed.  pr.  1598.  Freiburg  in  Uchtland,  1598. 

A  Freiburg  Recheiimeister  of  the  latter  half  of  the  sixteenth  century. 

Title.    See  Fig.  205. 


in  S|en)C0 

Dm  anfahcitDcrt  Arithmeucts  §ug<brau.' 

<j5en  M)\  t©KJl(fc§/mit  gar  f  ur^cn  prcceptt/ 
(edd(eic^mt>o2ma(0nie 

3urc^  3o§anncm  JriDolmum  fauten^ 


I        Magtllro  VVilhelmo  Mxfs.  Anno 


1598. 


Fig.  205.    Title  page  of  lautenschi 


PRINTED    BOOKS 


411 


Dcscnptioi.  8°,  9.4  X  14.7  cm.,  the  text  Xmixw^  6.9  X  12.3  cm. 
4  pp.  unnumb.  +  58  numb.  =  62  \^\>  ,  21-30  II.  r>cibur<,^  in 
Uchtland,  1598. 


oDi'rdwlDcnCHcjjul* 

©cnannt  aurca  propcuionum 
;Oariim&  &a6  fTe  gat  teguenriicf) 
3m  2lauff/bfgert  ^rei[^ttIg/lMmlic^ 

©as  21?mb  foU  (?f  t  )?<*«»  »n  ^«  »™"«»* ' 
5)cr2lauffoo:n«»/Cje5M9  64in&«n/ 

.^itib<ti('»o:ucn  gUicl>  07ame«.nd)t  / 

^le  llnii  &  e  gro)";  all jdt  jerferlcfit  / 
O0?ult  plicier  6ie  ftin&cr  ool)I 

Wit  Ocr  mittlcrcji  aUema^fl 
Jficil  spromifttmit  CcmwortieritaB  / 

®o  fpm&t  D»r  bein  ^r^jg  ^"^  2(uffga& . 
1)erFch:^?iegultt)ifteem  prob  f^rt^er^r 

■ZUas  erj!Iicl)g|Ianl>cn  if?  babinDcn 
'3}?iir^  t»o:iicn^cfo:&et  fjinbcn  fte^cn/ 

e^.-fafit  muf;  m  &ie  mitten  gef>ctt/ 
^ic  mitlcr  oaljlrau^  fommcnfcU/ 

;j\om()t  jlc  f"?  (?^l^s  getroffcn  tv  Jl, 


C5  X       3«m 


Kk;.  206.    From   lautenschlager 


Editions.    There  was  no  other  edition. 

This  is  the  first  arithmetic  that   1  have  seen  composed  entirely  in 
rhyme.     Various    others    had    already    contained    verses,    but    this    is 


412  RARA   ARITHMETICA 

substantially  all  in  meter.  Like  all  such  educational  vagaries,  it  is  weak 
in  subject  matter,  the  most  difficult  example  in  division  being  that  of 
65443  by  6.    The  illustration   (Fig.   206)   shows  the  rule  of  three  in 


CASPAR  SCHLEUPNER.     Ed.  pr.  1598  Leipzig,  1598. 

A  Breslau  Rechenmeister,  born  at  Niirnberg  c.  1535. 

Title.    See  Fig.  207. 

Colophon.  'Gedrtickt  zu//  Leipzig/ bey  Frantz  //  Schnellboltz. 
//Typis  haeredum  Beyeri.//(Woodct;t.)  Im  lahr//  M.  D.  XCviij.' 
(F.  99,  V.) 

Description.  8°,  9.3  X  14.7  cm.,  the  te.xt  being  6.9  X  12.2  cm. 
99  ff.  unnumb.,  22-26  11.    Leipzig,  1598. 

Editions.  Leipzig,  1598,  8°  (here  described);  Breslau,  1599,8°. 

Schleupner  was  one  of  the  last  serious  advocates  of  the  old  '  line 
reckoning'  with  counters.  He  was  a  disciple  of  the  Newdorffer  school 
of  Niirnberg  Rechenmeisters,  as  he  states  in  his  preface,  and  had  him- 
self taught  at  '  Neyfe '  and  '  Brefzlavv '  for  many  years  when  he  decided 
to  write  this  book.  He  makes  an  attempt  at  an  easy  method  of  pre- 
senting the  fundamental  operations,  hoping,  as  he  says,  to  set  forth  the 
doctrines  of  Adam  Riese  and  Johan  Seckerwitz  (another  Niirnberg 
Rechenmeister)  in  simple  fashion.  The  work  is  made  up  of  a  series  of 
impossible  questions  and  answers  between  a  father  and  his  son.  The 
latter  always  begins  with  '  Geliebter  Vater,'  and  the  father's  replies, 
while  always  clear,  are  often  very  extended,  the  entire  99  pages  cover- 
ing little  more  than  the  four  operations,  together  with  a  few  insignifi- 
cant problems.  As  a  simple  presentation  of  line  reckoning,  however, 
the  book  has  few  equals. 

JOSEPHUS  UNICORNUS.  Ed.  pr.  1598.  Venice,  1598. 
See  p.  298. 

Title.    See  Fig.  208. 

'  Parte  Seconda,'  of  the  same  date,  follows  f.  204,  the  two  being 
bound  in  one,  and  the  pages  being  numbered  consecutively. 

Colophon.  '  In  Venetia,//  Apreffo  Francefco  de'  Francefchi 
Senefe.//M  D  XCVIIL'    (F.  413,  v.) 

Description.  4°,  i  5.5  X  21.6  cm.,  the  text  being  10. 4X  17.6  cm. 
12  ff.  unnumb.  +  395  numb.  =  407  ff.,  29-39  11.    Venice,  1598. 


PRINTED   BOOKS  413 


Un  btmn 

^oraug  citt  ficifs^cr  £efcr/  rte(p 

d)er  aCctne  fern  ^m  ma»<&in^  btf  aaff  i?ier  mat 

CQeune/auf^weiiDig  fan/  obne  eini^ec  CO?unbIicf)Cn  bc^ 

Ifidjt/  f»u'  fid)  felbff  /  bie  €inien  9vccftiumg  Icrnni  Ian/ 

IiUil3£fcl)a)ci5cn'bcr/  ivelc^cr  juu.or  fcfeott 

feme  (;5pccic8  aafftex  Jclicm  ge^ 

Ternet  ^at. 

Scrogleic^cn  lienor)  In  iDcntfc^cr 

©urc6  Safpar  ©cf jlcupncr/ 

von  TJilfmbcr^  /Scul>fcfjcn  ©cfjutmciV 

j](r  ju^rc^laii>.  'iDJennivjIictjcn/ vnD 

beucv  bet  Iiebc  n  ^o^cnti/  ju  nutj 

t)erfet(igct. 

3w  vorlesun^  2(n&rc^  <;Bo[cf  en  <5r6(n  /  i>n5 

Fig.  207.    Title  page  ok  schleupner 


LARITHMETICA 

VNIVERSALE 

DEL  SIG. 

lOSEPPO   VNICORNO, 

MATHEMATICO 
ECCELLENTISSIMO. 

Parte  Prima: 

Ndlaqualefi  conticnenon  folo  laTheoricadi  tutdi  Humeri. 

ma  ancoralaPratrica  appartenente  a 

tuminegotijhumani. 

trattata ,  &  amptifcatacon  fimma  eruditions  ,€  cm 
noui,  &  tfquifiti  modi  di  chiare^a^ 

CON    PRIVILEGIO. 


InVmtfa.apprefo  France/code  Frarjcefchi  //;^. 

Con  licen^^deSuperiori, 

Fig.  208.   Title  page  of  unicornus 


PRINTED    1500KS  415 

Editions.    There  was  no  other  edition. 

This  is  one  of  the  most  elaborate  treatises  on  arithmetic  published 
in  Italy  in  the  sixteenth  century.  It  consists  of  six  books,  the  first  four 
making  up  Part  I.  The  first  book  treats  in  a  detailed  fashion  of  the 
funtlamental  operations.  Unicornus,  for  example,  gives  six  methods  of 
multiplication,  a  treatment  that  recalls  those  of  Paciuolo  and  Tarta- 
glia.  There  is  a  good  discussion  of  the  two  general  methods  of  dividing, 
the  downward  ('a  danda')  method  having  as  much  attention  as  the 
galley  plan.  Fractions  are  also  treated  in  Book  I.  Book  II  deals  with 
the  theory  of  numbers  after  the  Boethian  method.  Book  III  treats  of 
roots,  surds,  and  proportion  ;  Book  IV,  of  the  rules  of  three  and  false ; 
and  Book  V,  of  business  arithmetic,  including  exchange,  interest,  and 
alligation. 

The  work  was  too  theoretical  to  be  popular,  but  it  is  an  excellent 
source  for  the  study  of  the  development  of  elementary  mathematics. 
Unicornus  gives  a  number  of  interesting  historical  references. 

Other  laorks  of  isqS.  Cassiodorus,  p.  2 1 1 ,  1 5 40  ;  Fischer  ( I^iscator), 
p.  247,  1549  ;  Moya,  p.  310,  1562  ;  Paolini,  p.  393,  1589  ;  Raymundus 
Lullius,  '  Opera  ea  quae  ad  adinventam  ab  ipso  artem  universalem  .  .  . 
pertinent  cum  diversorum  commentariis,'  Strasburg,  1598,  8°  (see  also 
p.  457);  Fabricio  Mordente,  '  Le  proposition!  di  Mordenti,'  Rome,  4° 
(geometric,  but  wdth  a  little  arithmetic). 

ANTHON  NEWDORFFER.     Ed.  pr.  1599.    Niirnberg,  i  599- 

One  of  the  famous  Newdoiffer  family  of  Nlirnberg  Rechenmeisters. 

Title.  '  Kunftliche  vnd  Ordentliche  An//\veyfzung  der  gantzen 
Practic  vff  //  den  Jetzigen  fchlag  vnd  derfelbenn  //  herlichcn 
gefchwinden  Exempel // vff s  kurtzt  ziifammen  getzogen  &c.// 
Meinen  lieben  Difcipeln  zu  fon-//derlichem  Niitzen  geftclt. 
Durch  //  mich  Anthonium  Newdorffer//Rechenmaifter  vnd  Mo- 
dift  der//Statt  Nurnberg// Anno  .M.D.IC  (F.  2,  r.)  On  f. 
I,  r.,  is  an  engraved  frontispiece  with  figures  of  Euclid  and  Py- 
thagoras, and  a  triangular  multiplication  table.  On  f.  i,  v.,  are 
14  11.  of  verse  'Ad  Stvdiosos  Artis  Nvmerandi.' 

ColopJion.  'Gedruckt  zu  Nurnberg/ //durch  Paulum  Kauff- 
mann.//M.  D.  XCIX.'     (F.  65,  r.) 

Description.  4°,  i  5  X  19.3  cm.,  the  text  being  1 1.4  X  15.5  cm. 
65  ff.  unnumb.,  22-32  11.    Nurnberg,  1599. 


4i6  RARA   ARITHMETICA 

Editions.    There  was  no  other  edition. 

This  is  an  excellent  illustration  of  the  work  of  the  celebrated  Niirn- 
berg  Rechenmeisters.  It  is  divided  into  twelve  books,  of  which  '  Das  erfte 
Buchlein  handelt  von  der  Venetianifchen  oder  Kauffmennifchen  Practic,' 
usually  called  Welsch  practice  by  the  German  writers.  It  is  composed 
chiefly  of  problems,  the  rules  and  explanations  being  left  for  the  master. 
Directions  are  given,  however,  in  the  case  of  fractions.  Book  II  relates 
to  denominate  numbers  and  the  '  Regula  de  Tribus.'  Book  III  treats  of 
the  weighing  of  commercial  products.  Book  IV  '  Handelt  von  der  Rech- 
nung  eines  Caffierers,'  including  exchange.  Book  V  is  entitled  '  lornates, 
Das  ift/  Rechnung  von  allerley  Handtierung,'  and  consists  of  practical 
problems  relating  to  the  purchase  of  goods.  Book  VI '  Tractiert  von  der 
Regel  Converfa  vnd  Quinque,'  inverse  and  compound  proportion.  The 
rest  of  the  twelve  books  are  devoted  to  practical  business  questions  of 
the  day. 

OBERTO  CANTONE.     Ed.  pr.  1599.  Naples,  1599. 

A  Genoese  arithmetician  of  the  sixteenth  century,  residing  in  Naples. 

Title.    See  Fig.  209. 

Description.  FoL,  14.5  X  20.1  cm.,  the  text  being  10.4  X  157 
cm.     304  pp.  (292  numb.),  34-3/  H-    Naples,  1599. 

Editions.  This  is  the  first  edition  of  this  work,  three  other 
editions  appearing  in  the  seventeenth  century. 

Arithmetics  written  by  Genoese  masters  in  the  sixteenth  century 
are  rare,  even  in  manuscript  form,  and  this  is  probably  the  only  such 
work  printed  in  Naples.  Genoa  was  a  mercantile  center,  but  its  dialect 
was  not  conducive  to  the  success  of  a  textbook,  as  is  seen  in  the  case 
of  Zucchetti's  treatise  (p.  425).  Naples  was  too  far  from  the  path  of 
international  commerce  to  produce  many  mercantile  works.  But 
although  Oberto  Cantone  was  a  Genoese,  he  was  a  '  professor  delle  dis- 
cipline matematiche  '  in  Naples,  and  he  dates  his  dedicatory  epistle 
'  Di  Napoli  li  15.  luglio  1599.'  Naples,  however,  had  a  commerce  of 
its  own,  even  if  not  as  extensive  as  that  of  Venice  or  Florence,  and  by 
the  close  of  the  century  it  was  natural  to  expect  works  of  this  kind. 

The  book  is  mercantile  and  is  based  upon  Borghi  and  similar  writers 
of  the  North.  It  gives  our  present  method  of  multiplication,  but  makes 
no  use  of  the  Venetian  or  Florentine  names,  with  the  exception  of  '  per 
colonna.'  Two  or  three  short  methods  are  given,  but  no  such  extended 
treatment  of  varied  forms  appears  as  in  Tartaglia,  or  his  great  prede- 
cessor Paciuolo.    Division,  except  in  simple  cases,  is  postponed  to  p.  142, 


L  VSO  PRATTICO 

DELUARITMETICA 

OBERTO  CANTONE 

DAG  ENO  VA, 

PROFESSOR  DELLE  DISCIPLINE 

Macematiche. 

n^EL  QVALE  CON  NVOVA  INVENTIONE 

/in/e^na  tn  materia  di  conti ,  I'vfe  tanto  del  Li  Regia  Camera  dtUa 

iSommaria,  quanta  di  Negottantiy  MercadantitC^Jrtegiani. 

€  come  Napoli  cambyi  €  recambij  in  ciajcunapazza, 

CON    PRIVILEGIO. 


o 


I  T^    7^.4"  V  0  L  ly 
^ppreffo  T  A  K  Qiy  I  til  o  Long  ho.     M.  D.  IC, 
Si  vendono  dal  ir.edcfiiriO  Aucorc  a  Banchi  nuoui. 
Fig,  209.   Title  fagk  of  cantone 


P 
< 

Q 

en 


4i8  RARA   ARITHMETICA 

and  is  there  treated  in  the  modern  form,  '  a  danda.'  Most  of  the  applied 
problems  are  in  exchange,  although  a  few  other  types  are  given.  The 
book  is  poorly  constructed,  being  too  prolix  in  the  treatment  of  the 
operations,  and  too  narrow  in  its  applications. 

JOANNES  MARIANA.    Ed.  pr.  1599.  Toledo,  1599. 

Born  at  Talavera  de  la  Reina,  in   1536;  died  at  Toledo,   February  17, 
1624.    He  was  a  Spanish  Jesuit  and  a  famous  historian. 

Title.    See  Fig.  210. 

Colophon.  ' Toleti,  Apud  Thomam  Guf-//manium,  Anno .  i  599.' 
(P.  206.) 

Description.  4°,i4X  18. 6cm., the  text  being  9. 3X  15.6cm.  6  pp. 
blank  +  8  unnumb.  +  192  numb.  =  206  pp.,  26  11.    Toledo,  1599. 

Editions.    There  was  no  other  edition. 

The  author  is  not  the  same  as  the  Giovanni  Mariani  mentioned  on 
p.  180,  but  a  Spanish  Jesuit  of  some  fifty  years  later.  The  work  is  on 
the  history  of  the  weights  and  measures  used  in  Spain  in  the  sixteenth 
century.  It  traces  these  measures  from  the  Roman,  Greek,  and  Hebrew 
sources,  and  is  valuable  for  the  study  of  the  history  of  the  subject.  It 
closes  with  a  table  of  comparative  measures. 

MARTIN  WENCESLAUS.    Ed.  pr.  1599.    Middelburg,  1599. 

A  Dutch  arithmetician  of  the  sixteenth  century. 

Title.    See  Fig.  211. 

Colophon.  '  Eynde  defes  onfes  eerften  Vo-//lumens :  Ghedruct 
int  Jaer  ons  Hee-/Aen  1599.  In  defe  vermaerde  Coop-/Atadt 
van  Middleburgh  in  //  Zeelandt.'// Also  in  French  :  '  Fin  de  ce 
noftre  premier  Volu-//men:  Imprime  I'An  de  noftre  Seig-//neur 
1 599.  A  la  tref-renommee  //  ville  Marchande  de  Middel-//bourg 
en  Zeelande.'    (F.  157,  r.) 

Description.  4°,  15.2  x  20.1  cm.,  printed  in  double  columns, 
one  in  Dutch  and  the  other  in  French,  the  Dutch  being  ^.J 
X  15.7  cm.,  the  French  5  X  15  cm.  4  ff.  blank  +  18  unnumb. 
+  137  numb.  =  159  ff.  Dutch:  30-41  II.;  French:  16-26  11. 
Middelburg,  1599. 

Editions.  This  is  evidently  the  first  edition,  since  the  dedica- 
tion is  dated  'Den  10.  dach  van  Decem.  ftijlo  nouo.  Anno  Chrifti. 


PRINTED    BOOKS 


419 


JO  ^'Hn  IS 

MARIANAE 

Hifpani. 

E  SOCIE.     lESV, 

DE   P  O  N'D  E  Rl'BV  S  ET 

menfuris. 


uimo 


^W^ 


CVM  PRIVILEGIO. 

Fig.  210.    Title  page  oi'  makiana 


420  RARA   ARITHMETICA 

T/FONDAMEKT 

B^n  ^nt\mttm:mtttt%w 

imntcht  J5?acti)tlt  /  nittitrsaDera  DalUt 

noonnmDuPr  ftuthm  bmi&mUrsgdban 

Smettil. 

:j5£p&t0ln  ji^cbrtDupt0cntietn  jfrancj^ots/ 

met  rcDelicUe  ouerccnacnmung^e  ofte 

XllCff 

^C0^   MARTINVM    VVENCESLAVM, 
AQVISGRANENSEM. 

LE    FONDEMENT 

DE  L'ARITHMETIQYE  AVEC 

LAPR.ACT1CQVE    ITALIENNE    E  N  S  E  M. 

blelcs  pieces ,  les  plus  necefsairesde  la  R.cglc 
d 'Inter  eft. 

ENSEMBLE  EN  BAS  ALLEMAN  ET  EN 
Jran^ois ,  avcc  raifonnable  Harmonie  on 
Concoidancc. 

IE  TOVT 

Pan  C^fJKTIir^i^rM   WE  NCE  S  LArM, 


MIDDELBVRGH^ 

^ji  SymoK  mroulm,  bjoonrriDnntit^^ucltenir.  1599, 

Cntft  menljintfe  te  tooge/  bp  Adiiaen  Dan  be  Vivere .  Tdotth'otuoopct/ 
Ujoonenbe  bp  be  nicuUJc  72>mfc/int>tn  bcrgulbcn  23pbeU 

Fig.  211.    Title  page  of  wenxeslaus 


PRINTl-:i)    IJOOKS  421 

1598,'  'Ic  10.  lour  dc  Dcccmhrc  ll\  lo  nouo.  Anno.  Chrisli-  1598.' 
The  long  and  stupid  preface  is  tlaled  November  30  of  the  same 
year.  Wenceslaus  had  already  i)ublished  two  works  before  this 
one,  the  '  Proportionale  ghesolveerde  Tafelen  van  Interest,'  i  594, 
8°,  and  the  '  Boukhoudens  Instruction,'  1595. 

The  book  is  interesting,  both  because  of  its  arrangcnienl  of  the  Dutch 
and  French  texts  in  parallel  columns,  thus  serving  a  purpose  in  lan- 
guage teaching  in  a  bilingual  country,  and  because  of  the  mercantile 
problems  which  reveal,  as  is  particularly  true  of  the  Dutch  books,  the 
contemporary  life  of  the  people. 

Other  works  of  IS99-  St.  Augustine  (see  Reisch),  p.  82,  1503  ; 
Barlaamo,  p.  343,  1572;  Benedetti,  p.  364,  1580;  Boethius  (see 
Reisch),  p.  82,  1503;  Bungus,  p.  384,  1583-84;  Capella,  p.  66, 
1499  ;  Clichtoveus  (see  Reisch),  p.  82,  1503;  Faber  Stapulensis  (see 
Reisch),  p.  82,  1503;  Finaeus  (see  Reisch),  p.  82,  1503;  Gallucci 
(see  Reisch),  p.  82,  1503;  Jacob,  p.  298,  1560;  Jordanus  (see 
Reisch),  p.  82,  1503;  Ramus,  p.  330,  1569;  Reisch,  p.  82,  1503; 
Schleupner,  p.  412,  1598;  Striibe,  p.  391,  1588;  Johann  Heere, 
'  Rechenbiichlein  von  allerhand  gebraiichlichen  Fragen,'  Niirnberg, 
8° ;  Andreas  Reinhard,  '  Drey  Regifter  Arithmetifcher  Anfang  zur 
Practik,'  Leipzig,  8°,  with  a  second  edition  in  1600. 

JACOB  VANDER  SCIIUERE. 

Ed.  pr.  1600.  Haarlem,  1600. 

A  Dutch  arithmetician  of  Meenen,  c.  1 550-1 620. 

Tit/c.    See  Fig.  212. 

Description.  8°,  9.4  X  14.4  cm.,  the  text  being  7.2  x  12.6  cm. 
2  ff.  unnumb.  +  202  numb.  =  204  ff  ,  22-23  l^-     Haarlem,  1600. 

Editions.  That  the  date  of  the  first  edition  is  1600  appears 
in  the  Voor-Reden  of  the  1625  edition,  where  his  son,  Denys, 
says  that  the  book  was  published  by  the  father  '  eerst  in't  laer 
1600.  ende  dit  is  al  de  vierde  mael  dat  het  gedrukt  is.'  There 
were  various  editions  in  the  seventeenth  century,  including 
the  following:  Haarlem,  161 1  ;  Rotterdam-Schiedam,  1624,  8° 
(p.  423) ;  Haarlem,  1625  ;  c.  1630,  8°  (p.  423)  ;  Gouda,  1634,  8° 
(p.  424);  Amsterdam,  1643,  8°  (p.  424);  Rotterdam,  1653, 
8°  (p.  425)  ;  Amsterdam,   1675. 


RARA   ARITHMETICA 


ARITHMETICA, 

Bttt^itumtt  Mtti  rc^oone 

<L^empeIen/ftec  nut  ttoo?  alle  Cooti^ 

lUtm/  fmtmfi/  CaffietiS  /  <®ntfan* 
lAQVES  VAN  DER  SCHVERE 

VAN    MeENEN. 

Nutei:  rildtFrancoyfche  School-meeftcr 

tot  Haerlkm. 


a5p  ^im  aooman  ©oetfebjucfeet/f  tt  te  Slaw- 

69ne-fliatt/iniKbCTSul0e|&atf?if,  i^oo. 

Fig.  2  12.   Title  page  of  vander  schuere 


PRINTED    HOOKS  423 

Vander  Schucre's  work  on  bookkecpinf^  is  mentioned  under 
the  later  editions  (p.  424),  although  it  was  not  published  in  the 
sixteenth  century.  No  effort  has  here  been  made  to  complete 
the  list  of  seventeenth-century  editions. 

This  is  one  of  the  many  practical  arithmetics  that  appeared  in 
Holland  about  this  time.  It  takes  up  the  fundamental  ])rocesses,  rule 
of  three,  fractions,  the  rule  of  practice,  partnership,  commissions,  in- 
heritance problems,  profit  and  loss,  interest,  exchange,  barter,  alliga- 
tion, and  the  various  other  rules  in  use  at  that  time.  It  was  one  of  the 
most  successful  Dutch  textbooks. 

JACOB  VANDER  SCIIUERE. 

Ed.  pr.  1600.  Rotterdam-Schiedam,  1624. 

See  p.  421. 

Title.  'Arithmetica//Oft//  Reken-konft;//Verciert  met  veel 
fchoone //  Exempelen/ zeer  nut  voor  alle  vlijtighe // Oeffenaers 
cnde  leer-ghierigheAenvanghers //defer  Konft/  etc.  Ghemaeckt 
door  //  lacob  Vander  Schuere,  Meenenaer,//  Nu  tcr  tijdt  Fran- 
foyfche.  SchooI-meefter//tot  Haerlem.//Ende  nu  int  hcrdrucken 
overfien  ende  ghebetert.// (Woodcut  of  Vander  Schuere  with 
motto:  Door  siet  den  Grond.)//Tot  Rotterdam,// Voor  Pieter 
van  Waefberghe/  //Anno  1642.'    (F.  i,  r.) 

Colophon.  'Tot  Schiedam, //Ghedruckt  by  Adriaen  van  Delf .// 
Anno  1624.'    (F.  208,  r.) 

Description.  8°,  8.7  X  14.7  cm.,  the  text  being  y .6  X  12.9 
cm.  2  ff.  unnumb.-f  206  numb.  =  208  ff.,  22-27  11-  Rotterdam- 
Schiedam,  1624. 

See  above. 

JACOB  VANDER  SCIIUERE. 

Ed.  pr.  1600.  S.  1.  a.  (.?),  c.  1630. 

See  p.  421. 

Title.    '  Arithmetica  oft  Rekcn-konst.' 

This  edition  is  without  date,  the  title  page  being  missing,  but 
isprobably  c.  1630.  F.  i  is  missing.  9  x  15.6  cm.,  the  text  being 
7.3  X  12.2  cm.     5  ff.  unnumb.  -F  252  numb.  =  257  ff.,  35  11. 


424  RARA   ARITHMETICA 

JACOB  VANDER  SCHUERE.    Ed.  pr.  1600.    Gouda,  1634. 

See  p.  421. 

Title.  'Arithmetica// oft //Reken-konft. //Door  lakob  vander 
Schuere,  Meenenaer  //  Eertyts  Frangoysche  Schoolmeefter//tot 
Haerlem,//En  in't  herdrucken  by  hem  ouerfien // verbetert  en 
vermeerdert,  en  noch  by-//geuoecht  een  kort  onderricht  van  // 
'titaliaens  Boeck-houden // ter  Goude // By  Pieter  Rammefeyn, 
Boeck-verkooper  //inde  Korte  Groenen-dal,  in't  Vergult  A,  B,  C. 
//ao  1634.  W  Akerfl :  fee'  (The  title  page  is  elaborately  en- 
graved, with  a  portrait  of  the  author  surrounded  by  the  following: 
'Doorseit  den  Grond  lakob  Vander  Schvere  aet  50.')    (F.  i,  r.) 

Description.  8°,  8.8  X  14.2  cm.,  the  text  being  7  x  12.7  cm. 
8  ff.  unnumb.  +  208  numb.  =  216  ff.  Bound  with  this  is  '  Kort 
onderricht  //over  het  //  Italiaens  //  Boek-houden ; //  Nu  int  licht 
ghebracht  //  Door  lakob  vander  Schvere,'  etc.,  with  i  f.  unnumb. 
+  37  numb.  =  38  ff.,  making  a  total  of  254  ff.  in  the  book, 
28-33  11-    Gouda,  1634. 

See  p.  423. 

JACOB  VANDER  SCHUERE. 

Ed.  pr.  1600.  Amsterdam,  1643. 

See  p.  421. 

Title.  'Arithmetica //oft  //  Reken-konft.//  Eneen  kort  onder- 
richt van't  Italiaens  Boeckhoude // Door  lacob  vander  Schvere 
Meenenaer.  //  By  den  Autheur  overfien,  verbeetert  en  vermeer- 
dert.//Tot  Amsterdam, //voor  Michiel  de  Groot  Boeckverkooper 
//  op  de  nieuwendyk  inde  Bieftkens  Bybel.  1643.'  (Engraving  of 
author  with  the  following  wording  :  '  lacob  vander  Schvere  Mee- 
nenaer. Out  67. laer.  Doorsiet  denGrondt.  Annoi643.')  (F.  i,r.) 

Description.  8°,  9.5  X  14.7  cm.,  the  text  being  7.2  X  12.5  cm. 
272  ff.  (216  numb  +  8  unnumb.  in  the  arithmetic),  26-31  11. 
Amsterdam,  1643.  Bound  with  this  is  '  Kort  onder-richt  //Over 
het  //Itahaens  //  Boeck-houden.//  In't  Licht  gebracht  //  door  // 
Jacob  van  der  Schuere.// t'Amsteldam,// By  Michiel  de  Groot, 


PRINTl^I)    HOOKS  425 

Boeckverkooper  //oj)  clc  Nicuwcndijck/tulTchcn  dc  twee  Ilaer-// 
lemmer  Sluyfcn.    A".  1675.'    (F.  i,  r.,  of  the  bookkeeping.) 
See  p.  423- 

JACOB  VANDER  SCMUERE. 

Ed  pr.  1600.  Rotterdam,  1653. 

See  p.  421. 

Title.  '  Arithmetica  //  ofte  //  Reken-konft, //  En  ecn  kort 
onder-richt  van't  Italiaens // Boeck-hoiiden.// Door // lacob  van 
der  Schuere  Meencnaer.//  By  tlen  Autheur  over-ficn,  verbetert 
en  //  vermeerdert.//Tot  Rotterdam, //Gedruckt  by  Pieter  Wacf- 
bergc,  woo-//nende  op't  Steyger/  in  de  gekroonde  //  Leeuw/ 
Anno  1653.'     (F.  i,  r.) 

Description.  8°,  8.7  X  14.  i  cm.,  the  text  being  7  x  12.1  cm. 
275  pp.  (219  numb.  +  8  unnumb.  in  the  arithmetic),  26-33  11- 
Rotterdam,  1653.  Bound  with  this  is  'Kort  onder-richt // over 
het  //  ItaHaens  Boeck-houden.  //  In't  Licht  gebracht  //  door  // 
Jacob  van  der  Schuere.// Tot  Rotterdam,// Gedruckt  by  Pieter 
van  Waefberge,//Ordinaris  Drucker/ woonende  op't  Steyger// 
in  de  gekroonde  Leeuw/ Anno  1653.'  (F.  i,  r.,  of  the  book- 
keeping.) 

See  p.  423. 

GIOVANNI  BATTISTA  ZUCHETTA. 

Ed.  pr.  1600.  Brescia,  1600. 

Born  April  21,  1550.     A  Genoese  arithmetician. 

Title.  See  Fig.  213.  The  privilege  is  dated  '  In  Genoua  nel 
di  30.  di  Genaro.    M.    CCCCCC 

Description.  FoL,  24.2  X  33.4  cm.,  the  text  being  17.  i  X  26 
cm.     444  pp.  (412  numb.),  33-37  11.     Brescia,  1600. 

Editions.  There  was  no  other  edition.  Brunet  says  this  '  Prima 
Parte '  is  the  only  one  that  appeared. 

The  work  has  several  interesting  features,  not  the  least  one  being  the 
apology  '  Al  generoso  lettore,'  in  which  the  author  speaks  of  the  criticism 
liable  to  be  directed  against  a  Genoese  author  on  account  of  his  pro- 
vincial Italian.    The  '  Prologo '  is  a  curious  dissertation  on  the  '  Arti, 


426 


RARA   ARITHMETICA 


Fig.  213.   Title  page  of  zuchetta 


PRINTED   BOOKS  427 

Scienze,  &  altro,'  with  some  ninety-eight  arguments  to  show  the  need 
for  arithmetic  on  the  part  of  all  classes  of  humanity.  The  farmer,  the 
musician,  the  thief,  the  cook,  the  prelate,  all  are  shown  to  have  need  of 
number;  and  Nature,  Intelligence,  and  even  God  himself  make  use  of  it. 
The  book  presupposes  a  knowledge  of  the  arithmetic  of  integers, 
and  opens  with  a  treatment  of  fractions.  The  rule  of  three,  in  all  of 
its  forms,  and  with  most  unbusinesslike  numbers,  is  then  discussed  at 
great  length,  and  this  is  followed  by  various  complications  of  the  Regola 
del  Cattaino,  '  cosi  detta  da  gli  Arabi  inuentori  di  quello,  ch'  in  lingua 
nostra  fignifica  falfa  pofizione.'  The  latter  part  of  the  book  treats  of 
such  topics  as  partnership,  barter,  and  alligation.  The  work  was  not  of 
a  nature  to  have  any  influence  on  Italian  arithmetic. 

WILHELM  SCHEY.     Ed.  pr.  1600.  Basel,  iCoo. 

A  German  Rechenmeister  at  Solothurn.    Born  c.  1560. 

Title.    See  Fig.  214. 

Description.    4°,  15.1  X  19.4  cm.,  the  text  being  10  X  15  cm. 
486  pp.  (470  numb.),  36-37  11.     Basel,  1600. 

Editions.   That  this  is  the  first  edition  appears  from  the  dedi- 
cation :  'Datum  den  12.  Septembris.     Anno  Domini  1600.' 

The  book  is  an  attempt  at  a  complete  commercial  arithmetic,  but  is 
not  well  constructed.  The  author  likes  to  arrange  his  computations  in 
a  bizarre  fashion,  for  effect.  He  extends  his  explanations  unduly,  and 
for  so  large  a  book  the  mercantile  information  is  not  as  complete  as  it 
should  be.  There  is  a  curious  arrangement  of  the  figures 
in  the  proofs  by  casting  out  nines,  as  here  shown,  although 
whether  this  is  a  typographical  matter  or  a  notion  of  Schey's 
there  is  nothing  to  indicate.  In  the  *  Re'gula  Falsi '  there  is 
a  rather  early  use  of  -^  for  the  minus  sign.  In  general  the 
book  is  reactionary,  giving  only  the  galley  division  (with 
much  attempt  at  effect),  and  mentioning,  although  not  treating, 
'  Duplatio  '  and  '  Mediatio.' 

Other  works  of  1600.  Barlaamo,  p.  343,  1572  ;  Buscher,  p.  393, 
1590;  Chambers  (see  Barlaamo),  p.  343,  1572  ;  Herodianus,  p.  60, 
1495  ;  Hylles,  p.  396,  1592  ;  Jacob,  p.  298,  1560  ;  Lonicerus,  p.  253, 
1551  ;  Ramus,  p.  263,  1555  ;  Reisch,  p.  82,  1503  ;  Reinhard,  p.  421, 
1599;  Schulze,  p.  383,  1584;  M.  van  den  Dycke,  '  Chyfer-Boeck  ' 
(second  edition,  I  do  not  know  the  first),  Antwerp,  8°,  and  '  La  vraye 
reigle  d'Arithm^tique '  (a  translation  of  the  other?),  ib.,  8°;  Georg 
Gleitsmann,  '  Kiinftlirhes  Rechenbuch  fowohl  auf  Linicn  als  mit  /.iffern 


428 


RARA   ARITHMETICA 


ARITMETICA 

U^ubc»/ittma^tn/iM  tin  |c&<r|tm(tc^c0  Vn^anbta/ 
(it  9W  ^unMid^en  ^erid^t/bon  im  itlbs/wet 

?)urc^  2Btl§dmum6c|^9/  5:eut^ 

f^n  ©c^wl  ^"«^  9?cc^ctttiiciffer  j«  @olo^ 

n<n  anfafxnDm  @cl)ul<rn/»n>  mm  K&m  ^egi^tidm 
Un&et/t>n^  in  Xruct  txtfctn^ct. 


Anno  M.  DC. 


P 


Fig.  214.   Title  page  of  schey 


PRINTED    BOOKS  429 

nach  defs  Rami  Arithmetica  geftellt,'  Frankfort,  8°;  M.  Johann  Taf, 
'  Schones  neues  .  .  .  Rechenbuch,'  Cologne,  4°.  There  was  also  pub- 
lished, s.  a.,  but  c.  1600,  a  work  by  Vincenzo  da  Bergamo,  '  Arith- 
metiche  instruttioni.' 

There  were  many  other  arithmetics  published  in  the  sixteenth  cen- 
tury without  date,  including  the  following  :  Anonymous, '  Art  et  Science 
de  Arismetique,'  Paris,  12°,  96  ff.  (Boncompagni  sale);  Anonymous, 
*  LivTC  des  gects,'  s.  1.,  4°  (see  p.  130,  1520)  ;  Anonymous,  '  Abbaco  di 
succincte  dimostrazioni,'  s.  1.  (Milan?)  (Brunet)  ;  Anonymous,  '  (Q)ui 
apresso  e  inaci  col  nome  di  dio  intedo  di  tractare  e  scriuere  alquatimo 
di  e  regole  sopra  larte  del  numero  altrimeti  chiamato  algurismo,'  s.  1., 
33  ff.  (Riccardi)  ;  Joachim  Ammonius,  '  Isagoge  Arithmetices  .  .  .  cum 
praefatione  P.  Melanchthoni,'  Wittenberg;  Angelus  Mutinens,  p.  140, 
1525  ;  Sarafino  da  Campora,  '  Delia  ragione  dell'  Abbaco,'  and  a  work 
on  the  calendar  (Messina,  1559,  and  Rome,  1560)  (Riccardi);  Lauro 
Quirini,  '  De  mysterio  numerorum  ;  '  Matteo  Ricci,  '  T'ung-wen  suan- 
chih  '  (Practical  arithmetic  in  1 1  books,  the  work  of  a  Jesuit  missionary 
in  China,  b.  1552,  d.  1610;  possibly  not  published  during  his  lifetime). 


PART  II 
MANUSCRIPTS 


?tmo«mic  tml^ir.ai^tiimttciDera 
ttiZa^tmiao  T<53;tittmtitC5fli45  «f 


\.      KkoM 


TA.Ml'ANUS    MANISCKII 


MANUSCRIPTS 


EUCLID.  Latin  MS.,  c.  1260. 

See  p.  II. 

Title.  'In  hoc  libro  otinet'  gcomct'a  cuclklis //cu  .:mento 
magri  campani.'    (F.   i,  v.) 

Colophon.  '  CExplicit  geometria  euclidis  cu  comcn//to  magiftri 
campani.'    (F.  165,  v.) 

Description.  Fol.,  17.8  x  25.8  cm.,  the  text  Ix-ing  8.5  x  17.2 
cm.  besides  the  marginal  figures.  165  ff.,  23-44  H-  ^^'I'itten  on 
vellum  c.  1260. 

Editions.  This  work  being  primarily  a  geometry,  I  have  not 
given  a  list  of  the  editions.  For  the  arithmetical  books,  which 
(except  for  Book  V,  on  proportion)  are  not  included  in  this 
manuscript,  see  p.  237. 

This  is  a  Latin  manuscri])t  of  Euclid,  with  the  commentary  (proofs) 
of  Campanus,  written  on  vellum  about  1260.  The  translation  of  the 
theorems  is  that  made  by  Adelard  (.Tthelhard)  of  Bath,  c.  1 120,  but  in 
the  early  printed  editions  of  the  '  Elements  '  it  is  generally  referred  to 
as  that  of  Campanus. 

Of  Campanus  himself  not  very  much  is  known.  His  first  name  was 
probably  Johannes,  and  he  is  known  to  have  prepared  a  set  of  planet- 
ary tables,  and  to  have  been  chaplain  to  Urban  IV.  (See  Boncompagni's 
Bulled  no,  I,  5,  and  XIX,  591.) 

This  manuscript  has  been  studied  by  Mr.  C.  S.  Peirce  {.Seieiiee,  x.s., 
XIII,  809),  who  believes  it  to  be  the  copy  given  by  Campanus  himself 
to  Jacques  Pantaloon  when  the  latter  was  Patriarch  of  Jerusalem,  hence 
before  August  29,  1261,  when  he  became  Pope  Urban  IV.  He  bases 
his  belief  on  a  sentence  written  in  a  cursive  hand  just  below  the  colo- 
phon, containing  the  words  'Jacobus  Dei  gratia  Patriarcha  Jerusalemi- 
torum.'    The  complete  sentence  is  :  'In  noie  dm  ame  Jacobus  del  g'' 

433 


434  RARA   ARITHMETICA 

patriarcha  Jerufalemitar  omibus  xpi  fidelibus  falutem  defiderium,'  — '  In 
the  name  of  the  Lord,  amen;  Jacob  by  the  grace  of  God  patriarch 
of  the  Jerusalemites  to  all  the  faithful  of  Christ,  greeting  and  love.' 
While  this  seems  more  of  a  blessing  or  quasi  imprimatur  than  a  mark 
of  ownership,  it  is  equally  valuable  in  serving  to  iix  the  date.  On  f .  i ,  r., 
is  the  inscription  in  a  fourteenth-century  hand,  'mgri  adolphi  di  Werda,' 
and  a  statement  that  the  manuscript  belonged  to  the  Phillips  collection, 
no.  4633.  On  f.  165,  v.,  is  an  inscription  in  an  English  hand  of  c.  1400, 
'  lib^  ifte  fuit  Di  armachani,'  — '  This  book  belonged  to  Dominus  Arma- 
chanus.'     (For  a  page  of  this  MS.  see  Plate  IV.) 

ANICIUS    MANLIUS   SEVERINUS   BOETHIUS. 

Latin  MS.,  c.  1294. 

See  p.  25. 

Title.  The  manuscript  begins  :  '  Incipit  prolog  in  arithmetica 
boetii.'    (F.  i,  v.) 

Colophon.  '  Explict  arifme^^  //  boetij  ad  fymacu9  patriciuc.' 
(F.  28,  r.) 

Description.  FoL,  19.8  X  27  cm.,  in  double  columns,  each  being 
7  X  20.1  cm.  28  ff.,  41  11.  Written  on  vellum  c.  1294.  Bound 
with  the  Euclid  described  below  but  in  a  different  hand.  It  is 
beautifully  written  and  illuminated.  The  contemporary  pigskin 
binding  has  the  inscription,  'lib'  arifmetice  boecij.'  The  text 
is  practically  that  followed  by  the  Friedlein  edition  (Leipzig, 
1867).     (See  Plate  I.) 

Editions.    For  printed  editions  see  p.  27. 

See  p.  27. 
EUCLID.  Latin  MS.,  c.  1294. 

See  p.  II. 

Title.  The  work  begins  without  title  :  '  Punct^  eft  cui9  ps  ne.' 
(F.  29,  V.) 

Colophon.  '  Explicit  geometri  euclidis  cum  comto  campani.' 
(F.  Ill,  V.) 

Description.  Fob,  19.8  x  27  cm.,  the  text  being  17.5  X  19.5 
cm.     Ill  ff.  (the  first  28  ff.  being  the  arithmetic  of  Boethius 


MANUSCRIPTS  435 

described  on  p.  434),  44  11.  W'riUen  on  vellum,  e.  1294.  It  has 
beautifully  executed  figures  and  is  a  fine  specimen  of  the  work  of 
the  mediaeval  scribe.  It  is  written  in  a  different  hand  from  that 
of  the  Boethius  with  which  it  is  bound.  The  cover,  which  seems 
contemporary  with  the  manuscript,  has  the  number  cclxxxxiiij, 
possibly  for  the  date  1294,  the  M  being  omitted  as  is  (;flen  the 
case.    (See  Fig.  215  and  Plate  V.) 

ANICIUS    MANLIUS    SEVERINUS    BOETHIUS. 

Latin  MS.,  c.  1300. 

See  p.  25. 
Title.  The  manuscript  begins:  Tncipit  libr  arismetice  art .  .  .' 
Description.  Fol.,  13  X  18.6  cm.,  the  text  being  8.9  X  14. i 
cm.  37  ff.,  32-36  11.  Beautifully  written  on  vellum.  The  work 
is  complete,  and,  like  the  manuscript  described  on  p.  434,  this 
shows  a  text  very  similar  to  that  followed  by  the  Friedlein 
edition.  At  the  end  of  the  manuscript  are  two  folios  of  com- 
mentary, closing  with  two  almost  illegible  lines  containing  the 
words  '  Com  .  .  .  campani  .  .  . ,'  referring  to  the  commentary  of 
Campanus.  Roman  numerals  are  used  throughout  the  text, 
which  was  not  always  the  case  in  manuscripts  of  this  date.  (See 
Plate  VI.)  The  commentary,  which  seems  to  have  been  added 
about  a  century  later,  has  some  Hindu-Arabic  numerals. 

PAOLO  DAGOMARI.  Italian  MS.,  c.  1339. 

Paolo  dell'Abaco,  Paolo  Astrologo,  Paolo  Geometra,  Paolo  Aris- 
METRA,  Paul  ok  the  Abacus.  Bom  in  Prato,  c.  1281  ;  died  at  Florence 
in  1374,  or,  according  to  some  writers,  in  1365.  He  was  a  celebrated  Flor- 
entine arithmetician,  'geometra  grandissimo,  e  peritissimo  aritmetico  .  .  . 
diligenti.ssimo  osservatore  delle  Stelle,  e  del  movimento  de'cieli,'  as  Villani 
{I.e  Vite  iV  Uomini  ilustri  Fiorentini)  calls  him. 

Title.  '  Trattato  d'Abbaco,  d' Astrono-//mia,  e  di  segreti  natu- 
rali //e  medicinali.'     (F.  i,r.) 

Descriptio7i.  Fol.,  21.7  X  29.3  cm.,  the  text  being  15  X  21.4 
cm.  138  ff.  (7  blank),  32-35  11.,  clearly  written  on  paper,  c.  1339 
(possibly  copied  later). 


'}y 


436  RARA   ARITHMETICA 

'J2.*  '      "  -c1K#ft|^«r<i1m«r4'-6^»<f.^-^md&^t•-CIn^1ilc^4• 

'     ^  af  ciocmquo(pfefitnbni    :{.t.fac?k»«*«*^tmjicrf> 

-^■^    > — 1-^  v-t-= — ,  ^^    v>>.v^sa.-^ ?^         eeJPV«-tmpiR-qu«©cfiTO^ 

\ — ^—t  »tlPlQ*dO  CtOrClTr  Olt -^    -mJartniftti-^^lfanTmipVnnW 

»■  ^lU  ^mttd  ■     <^er^v-  vmoa  -a.tc-  >  ■  >J^lm^^^a*;v^f«8  eicif  tmunr' 

\^„^  l^»  ^^«  pr^jftir-  :jjSiw  IP  4lv}vu>  MiuWini-  mcSiifnrae  cm*!*  iw5i«« 

.     ^^dwr ^ «f- imfji". qju'tfl m'Kr-ftf.pfm ^Droino <rtr^^'  e>^uf  vrn* 

>2'(~.     ' — b — ^ '  -  Vimi*  ni?mcMetad^r^rn^^4»ratni»niiVtiri«n8  4ttVdtnmf 

/^'      I — j-i   ^—^ — I  -iW»-d'*^tiBjto^lr<^tbmjn'tn4SiRihTfmip 

ir^   t— » — ,  Cfb^tf  tn^m*--   /f^ft-fivUftra-eiraiitS-^Hpn-mkn^i-^micrd 

Jfrtft^tiVCfvJ^m ^^-r- <ti  t- *if ■c<iS -A- ^.<- m'uirt- fkraxpr-niW- J- citr 

/  -  ^      ^k_  4^uob»cu^m<*^«nei^l\r^fmarfal• 

I — - — 3 1  on  nwcncxae  tft-par  eft-   -facoipfttt^fin^^Tinpint-sircra 

( 1    ' ^ — •  ♦  __>      •f-Vmornifl  itqjifo  ifff  cfiftmkra- «li<iiu»ijTt5 

Fig.  215.    From  the   1294  manuscript  of  euclid 

Bernardino  Baldi  testifies  to  the  esteem  in  which  the  author  of  this 
work  was  held,  in  the  following  words  :  '  Di  patria  Fiorentino  fu  Pauolo  ; 
il  quale,  per  1'  eccellenza  ch'  egli  hebbe  ne  le  Matematiche,  lasciato  il 
proprio  cognome,  fu  chiamato  da  tutti  il  Geometra.  Come  apunto  fra 
gli  antichi  auenne  ad  Apollonio  Pergec' 

This  manuscript  is  primarily  a  treatise  on  arithmetic.  The  writer, 
however,  left  a  number  of  blank  pages  at  the  end,  and  these  have  been 
filled  in  from  time  to  time  by  various  owners. 


Hamii-iiiTTUir'iivrx-MvT'Vfl'^'"!.  )l'"'«-"llTiT  C;'iiii,irMolilofti(J5.iMqni.,il)nd  ,"'. 

iugl>^IVttml^ai^■>l.7ullllr  gK""""''""!-™'''  <■•'>'"  ""'jr,''""  O'l-i-  )thniotniijr''iniir 

"bS  /w-*  ;>v:niuiii«fiii  !ff1i'U<»-lL>''U*>'ni  •'  lHI^Ulum^tn^1^«Jn'>«ltnJII^frcllll  fimi; 
y^jlrtiurr.aiiiwqiiiniiiif  srIiU mrdi^  VMir.iiro«-a  (iituianff.^.ilevjfc.uijfvvumi 

•tKMiin.ivntr.idAiii-'tW  •lr»'iVtmiiiniiitfi<(r.r'iHAiMftj{oftn!!»''«f  H'^(fi'''«q 
>t<hixiii,ai?j*uar»iimitfnu-ii4i  .vV'i1jn.fuO'm"mrr.nilt^in<,oippUU>ftviti 
»aOiftOTulunbvmo.iqU'aMmilUuiqiM^v<nrT..}»-t'WWWl<£mnI.<vbi.a.v^^ 
icfia-vtaaawnlliiHit- irtmojA.) «f ■  ■i5»<-VMM-iiii5ri'"l-^" "11  •»"fr-l'U V  'i5-^''r.';<llt' 
mia'r'inra: iiii!amriiiiiiiai'«*Mtymui  <v. Iftfiii c<ii3ift»uiniori Urtu.,i i.  f  f\^\gut utojrii 
plto.^<f.Ali;^lU)Von/*')■ftlDll^^uHft^^^(^■«f^ll»— Anjr.r.irflinjjwnT-  jjir 

^;t<ttllt\■a^rlITir.^llou^■v»;t<rt^lOfllmllld^■tv^ll■Il)■y^'Sa&ll^-ll•rt^•J0*m 
<  1A.ihrJirur^"ai{>idbna(rir.jii«ulij>Ai-i)ir«7t4li»»iirj'iit>iiitri»r^»,'i»>inmu-  _ 
•  r^i«^^t^hl^a.U>.^lOlumu•cqllt1l\^■ll>ll^lr^la4l^■<i^•;^W^:«l'^^llJI  0<li>iihiuJ..i(7icrr  J 

ptno.uig'rvtti  ('iitnuuf  j>re.iv)ni-v*itti'iic  h^m'.HOOni  •  •  - 

incrim  ^ucto  d:tJA»inir  cnuc  C»uob?  ac>?(ino  riuc  cri»uoli; 

.  yii>rrTiuj(ir.iil'r-nii'(iii{p.i,in-rvrtul>' 

■i^{i«rro,<ir.fiiiii'On>[U)q'i.\Hitr>  ^tiiumJ  i>inctiiiiiii  ('ittvuniiin  i\i<i(«irui 

HlCT.l»lMW^■SchuTk^all•*•^-tn  vhTicrif-rrt1'<.q^'''5?nrirtTKUiiii>».i<v.<i\V(v^ 
■til  TigrticuiiraiWS.iiiic.jn'nquidiioWrTMaCinifrniMjt^j^iHUjr.AiiiB.t^i'JiioO'iujj- 
'&iiasaj'hiunutioot^iiiio^b^uifctTXTlinTOJmf^ppiniiijralKlOM.CT'iqiirTqii'Hioi) 
ntt^'iif;*lvi9f<re(hi.  /k-m-|;.j»C5.liiiiiiiJ.ic.li<M6iicoTr,l>lilfiiniui,  qitiwi;i>ii(<a  , 

«i%Mi'irai)*'trtt«e-ClCijf'fi-U'<^"'<\F'l"'^'^Wi''i<''''^'™'^^'lP';T''lhnaTlff^^ 

S*»     iJwmii.tajr-li-J'TOCv^if  nij}f.lYncimc«<viiTi:v'J<hir  frihiyrtrtoiiiiijfMl'r;^^^  ' 

<pqihtmitt=tfomoriMam*  ►Oiiiifiiui<>!<v>'ihm!('qfl?,lH\i'iilyj'aiieiiitofMii 
q»hn^lMllfJ^^^!^lUrI^'^l•*ll'^l^"<qluli^«<Jy^t^lli^lul^:tl^>^^da»n5t-aX■f^Ull■0>^^ 
m  'fCno<t.jjiVaa»TMiicA>.mtW^/Vur-  iwK'giliUniii'cjd^imiiii^irlili.vv  «i'lv)HiII«»ct<J). 

^Jq^'m),mlomn5{■laT^Illfe^ucto4^ll^fcauM't^SWy(te«(t^^ 
a  Aftuobi  nriinuie  Vitib\  Ccfinb^Drnxr  i«h  if  C  aurr  atiumie 

Tlrf  /»..»       '*^  "•  <V^^Jml»^lA•c■o0^^n^rl  •n'vrign,in-.ijirraiittto-.i?-rrir(vqtfini<m>" 
UIO  OftuUinn*- ar  oqlf  qCijmid>uo»iumi.iMUsvctvru„i-iiTch8.l>iro.uia^^^ 
l.mlrf\lr>!»l^OTIfc<rert^l,etIr«0lrt^^e^3l^•^U>_vttlOllHMl«b^^l^lnUilKf^•llUUl 
^«lu^jOTiov-qioawj>ima)luimOrairq;('j\trt*re.qi-«miiti»r.«qi^^ 

,  nm 4ujm (ham q v.- oirtH f^vw- f- l.i.vuf. qmqTni fuH-qu.i « <q-|cu-!nr.Nr,ninr      ^^ 
tiV9«njrli.«i,jFui.r.ctCTtiie-q-«ytorinTTfmfvnr'>jtit,j>^^7  | 

^,  :i[»e  V.UCUO  tonrtrnid^'UiMuiiuh 

^'i«trMJri4>f<u  rrf ^-r  i  mo  i>i  i({^  i 


Plate  V.    From  a  manuscript  of  euci.id,  c.  1294 


Vv 


MANUSCRIPTS  437 

The  arithmetic  begins  (f.  i,  r.)  :  '  VA  nome  sia  di  Dio  et  a  revcrentia 
//della  fuo  potentia  et  della  fanta  trinitade.  Kt  dello  fuo  madre // 
uirgino  fempre  fanta  maria  Et  del  beato  Sco  Giouani  batifto  //  .  .  .  Al 
chominiciamento  del  noftro  trattato  .  .  .'    On   f.   i,  r.,  is  a  table  of 

j^  j*<o  »vm«»  r^p-vtA^n^  I4^dt^»;  C^  cfJ>-^^p^^f*^t<r'^')r')■\^<■^S 
<  \bf  c  7^  cVc-jW-  -P^vwrrvo  r^r-t^m^*  ^'icr-^f^^Vc-w^r- AM»Vh)4«r-i<' 


V 


T>frm.M'  avtA»0»  ^-Vtixn^*lx»««^  i\*vn^*«l»  t'W>«'^^'>-'n<p'  c ;^/^^t(-^,■^o^fc-- 
»lc.o  t<*rm<.  vn<^u— vftn*-JX^  -L^'^n.-mvn  t'TJir/V-no  y  l^  »^>*«»°  vu^lc-lAt- 

Fig.  216.   From  the  dagomari  manuscrift 

money.  F.  2,  r.,  —  f.  3,  v.,  is  a  table  of  contents  :  '  Quefti  sono  echapitoli 
del  noftro  tractato,'  and  this  states  that  the  work  includes  the  '  Re- 
gholvzze  del  Maestro  Pagolo  Astrologo.'  (ff.  121,  r.,  —  131,  r.).  This 
part  of  the  volume  is  an  ordinary  commercial 
arithmetic  such  as  the  Florentine  teachers  pro- 
duced in  the  fourteenth  and  fifteenth  centuries. 
The  writing  and  the  forms  of  the  numerals  indi- 
cate this  period,  and  are  not  unlike  that  of  a 
fourteenth-century  computus  (see  p.  445)  in  this 
library.  This  may,  however,  be  a  fifteenth-century  copy.  The  exam- 
ples in  division  are  unique,  since  they  follow  neither  the  galley  nor  the 
'a  danda'  method,  as  is  here  shown  in  the  case  of  49289-5-23  =2143. 
The  peculiar  position  of  the  remainders  should  be  noted.  The  o  indi- 
cates no  remainder,  and  the  2,  2,  and  i  are  excesses  of  y's  in  the  proof. 


438 


KARA   ARITHMETICA 


That  the  author  considers  this  the  '  a  danda '  plan  (the  forerunner  of 
our  present  long  division)  appears  in  his  use  of  this  name  immediately 
after.    Another  odd  feature  is  the  placing  of  the  divisor  second  in  the 


ct»il|>ipv-£lh>«ron<- 

»linn«  pcpr— /<J/e»KMV-  n\--VWlnt-*ttt 

mc'Vc.y>o[V-/')/<mflV-/f\»>Ml».|>U«<^/-/9/vvi^1->-fU'.Ii~i«.5  •  cfAiiu{jf_-  f-i^pl*- 


rr' 


'Vl*^' 


•  -^>»>»w«^/  Srwti*!  pf<n(^ 


ll.4^>/-MSnc4^t4><jMAW-r-    f»nnc-t^>'  in^tmth>  fefa^Anhr^  net^f^n^ 


Fig.  217.   From  the  dagomari  manuscript 


division  of  fractions,  it  usually  appearing  first  in  that  period.  The 
problems  are  of  the  usual  Florentine  mercantile  character.  The  first 
few  are  followed  by  a  treatise  on  'Nvmeri  perfetti '  (f.  67,  r.),  and  this 


MANUSCRIPTS 

(f.  73,  r.)   by  further  business  i)rol)lenis.     The  wurk 
curious  illustrations  (Figs.  217,  218). 

The  most  interesting  feature  of  the  treatise  is, 
however,  the  internal  evidence  as  to  its  date.  It  is 
usually  possible  to  determine  (juite  accurately  the 
date  of  a  Florentine  manuscript  on  arithmetic  by 
the  examples  in  equation  of  payments,  a  favorite 
application  with  the  Tuscan  arithmeticians,  and  one 
requiring  the  year  to  be  stated.  This  is  the  case  here, 
where  the  dates  in  these  examples  are  all,  save  two, 
1339.  The  writer  has  also  used  approximately  these 
dates  in  other  examples,  in  part  as  follows:  1329- 
1332  (a  problem  on  the  calendar,  f.  27,  r.)  ;  1310- 
1404  (an  astronomical  table,  which  would  naturally 
extend  well  into  the  future,  f.  79,  r.)  ;  1330  (f.  i  i6,r.), 
and  1339  frequently.  A  table  of  the  19-year  cycle 
(f.  123,  V.)  begins  with  1337,  as  is  easily  computed 
from  a  marginal  note  in  a  different  hand  bearing  the 
dates  1394  and  141 2.  This  part  of  the  work  ends 
on  f.  131,  r.,  and  there  can  be  no  doubt  that  it  was 
written  about  1339,  the  date  so  freciuently  used  in 
the  problems. 

F.  1 2 1 ,  r.,  begins  '  Regholvzze  del  Maestro  Pagolo 
Astrologo,'  Dagomari  being  referred  to  in  the  table 
of  contents  as  Paul  of  the  Abacus,  '  Regholuze  del 
Maeftro  pagholo  delabacho  .  .  .'  The  '  Regholvzze  ' 
was  first  published  in  Libri's  Histoirc,  vol.  Ill,  p.  296. 

Dagomari  is  included  in  the  list  of  Bernardino 
Baldi's  (1589)  biographies,  published  in  the  Bon- 
compagni  BiiUctino,  XIX,  600. 

Ff.  131,  v.,  and  132^  r.,  are  in  a  different  hand. 
These  and  the  following  leaves  were  originally  blank, 
and  after  ff.  132,  v.,—  133,  v.,  had  been  written 
upon,  some  owner  used  these  two  blank  pages.  He 
has  also  left  his  dates,  viz.  1400,  1402,  1406,  14 12, 
1435,  the  1402  being  used  several  times.  This  was 
therefore  written  about  1 400-1 435.  In  this  occurs 
the  first  per-cent  sign  I  have  met,  other  than  p.  100, 
which  is  in  the  earlier  part  of  this  volume.  This 
unknown  writer  of  about  1425  uses  a  symbol  which, 
by  natural  stages,  developed  into  our  present   %. 


439 

contains  several 


P, 


Fk;.  21S.     From 

THE     DAGOMARI 
MANUSCRIPT 


440  RARA   ARITHMETICA 

Instead  of  writing  'per  loo,'  'p  loo,'  or  'p  cento,'  as  had  commonly 
been  done  before  him,  he  wrote  'p  ^'  for  '  p  c,'  just  as  the  Italians 
wrote  1,5,...  and  i°,  2°,  ...  for  primo,  secundo,  etc.  In  the  manu- 
scripts which  I  have  examined  the  evolution  is  easily  traced,  the  ^ 
becoming   ^  about   1650,  the  original  meaning  having  even  then  been 

• <^    8c  o 

Fig.  219.    From  the  addition  (c.  1400-1435)  to  dagomari 

lost.  Of  late  the  '  per'  has  been  dropped,  leaving  only  g  or  %.  See 
Figs.  219,  220. 

Ff.  132,  v.,  and  132,  r.,  are  in  a  different  hand.  They  refer  to  the 
calendar,  and  contain  the  dates  1380  and  1382,  each  twice.  F.  133, 
v.,  is  in  a  still  different  hand,  although  also  on  the  calendar.  The  date 
of  this  part  is  fixed  by  the  expression,  '  et  i  ifto  prefeti  aho.  f.  1447  finit 
circuluf  anni.' 

EUCLID.  Latin  MS.,  c.  1350. 

See  p.  II. 

Title.    None.     Fragments  of  the  '  Elements.' 

Description.  4°,  16.5  X  21.4  cm.,  the  text  being  10.5  X  15.8 
cm.,  with  marginal  drawings.  6ff.  unnumb.,  32-41  11.  A  Latin 
manuscript,  written  on  vellum  in  a  hand  of  c.  1350. 

The  manuscript  is  a  fragment  of  6  folios,  and  includes  part  of  Books 
III  and  IV,  and  a  list  of  propositions  evidently  based  upon,  but  not 
identical  with,  Euclid's  sequence. 


intf  a^ ai.'-\ iyiti.)i4f4--  wU- luK  ilifli.i; .^(c  ,»\e ,j,ii  im n,.,# ^.,„<^ .^y y.,n viVi 

liito  nn\1\»-'-"tti  nK\lioracrfo<a>mvi'lirU-nKliltw^ui  .]loiiouri'.«.riTi\tl  avf 
r^-.^'nif  4'  huim  M«Ti.Mi  (\\n  -»'d'''iccrii.vlUv  vltihtioit'iur.'n  f.iTil  if  i-  ij.Toinc 
re.lJp^ol•^«.^lIc:^(■utc111C.f^..rt1..lyvl1l.^.^^  Unwivul  co^citi  .S-t^i.K  »-v,nT».iW"- 
i^iMlCTj>ji»roo.« A -r c^ ■\mi<c'f\it>  ciTniitruiV  idttn .?. vt  ojiwi  fie fr mtilor 


w.y  ht  crdTio^-cicf  ig-.cmuict^  .rtirojcntt  tunxfy?.    ^^ -     ) 


»  t-  1 .  :;tl>— -J. 


it'i/iiif.^VV'iii. 


^-^> 


rc.Txr  a«plc  r.5«.H.U5Vcnfrnuriu.>.rii:faosf.jiviu.cV!l  tiouoiAnomMw 


"'Sk 


Plate  VI.  From  a  manlsckut  of  hoethius,  c.  1300 


MANUSCRIPTS  441 


-2-4-4^ 


66 


00 


o 


n.  cj  a0^  /a. 


Fig.   220.    From  a    1684   .manusckii'T,  not  hkkk  catal()(;iki) 


442  RARA   ARITHMETICA 

ALBERTUS  MAGNUS.  Latin  MS.,  c.  1350. 

Albertus  Teutonicus,  de  Colonia,  or  Ratisbonensis.  Bom  at  Lau- 
ingen,  Swabia,  c.  1 193  ;  died  at  Cologne,  November  25,  1280.  He  was  Count 
of  Bollstadt,  a  Dominican  priest,  and  Bishop  of  Regensburg.  He  studied  at 
Padua  and  taught  at  Bologna,  Strasburg,  Freiburg,  Cologne,  and  Paris.  He 
was  so  prominent  as  a  philosopher  that  he  was  known  as  '  Doctor  Universalis.' 

Title.    'De  C^lo  et  Mundo.'    (F.  i,  r.) 

Description.  22.4  X  32  cm.,  the  text  being  14  X  21.3  cm.  with- 
out the  marginal  notes,  and  arranged  in  double  columns,  each  6.5 
cm.  wide.    90  ff.,  50  11. 

This  is  a  beautifully  written  manuscript,  in  a  fourteenth-century 
hand,  with  fine  initials  in  red,  black,  and  blue.  The  first  folio  contains 
part  of  the  calendar,  and  a  few  random  memoranda,  including  an  old 
price  mark  of  three  ducats.  The  twenty-ninth  folio  is  blank  except  for 
some  crude  circles,  and  one  folio  has  been  cut  out.  There  is  a  break 
here  in  the  manuscript,  f.  28,  v.,  closing  with  these  words  from  the  fourth 
tractatus  of  liber  I  :  '  Rei  a'  gnate  e  ultilh  2  finis  in  tpr  aut  actu  aut  po. 
quia  si  e  corrupta  tiic  ht  actu  fine  2  si  e  adhuc  corrupt'.'  ('  Rei  autem 
generatse  est  ultimum  et  finis  in  tempore  aut  actu  aut  potentia  :  quia 
si  est  corrupta,  tunc  habet  actu  finem  :  et  si  est  adhuc  corruptabilis  '  — 
the  rest  of  the  sentence  being  '  habet  finem  potentia.')  After  the 
blank  folio,  f.  31,  r.,  opens  with  these  words  from  tractatus  I  of  liber  II : 
'  que  eft  f  b'a  sepata  '  ('  quae  est  substantia  separata,'  the  preceding  miss- 
ing words  of  the  sentence  being  '  quae  tamen  non  limitant  operationem 
formae  ejus  ')•  Aside  from  this  the  manuscript  is  complete.  F.  89  closes 
in  col.  I  with  the  words  '  Explicit  liber  de  celo  et  mildo  fratris  alberti  deo 
agamus  gratias.'    F.  90,  r.,  is  blank,  but  90,  v.,  has  the  zodiac  and  planets. 

There  is  a  brief  reference  to  Pythagorean  arithmetic  in  liber  I, 
tractatus  I,  caput  II. 

The  best  edition  of  this  work  is  in  '  B.  Alberti  Magni  //  Ratisbonensis 
Episcopi,ordinisPraedicatorum,// opera  omnia,//  .  .  .  cura  ac  labore // 
Augusti  Borgnet.// Volumen  quartum  //  Parisiis  .  .  .  MDCCCXC  The 
first  edition  of  the  Opera  Omnia  appeared  in  Leyden  in  165 1. 

EUCLID.  Latin  MS.,  1375. 

See  p.  II. 

Title.    The  work  begins  :   '  Functus  eft  cui^  ps  no  e~.' 
Description.    Fob,  20.9  X  30.1  cm.,  the  text  being  1 1.9  X  19.5 

cm.     39  ff.  unnumb.  -f  i  blank  =  40  ff.,  44-47  11.    A  Latin  man- 

uscri])t,  written  on  paper,  c.  1375. 


MANUSCRIPTS  443 

This  manuscript  includes  the  fust  five  hooks  of  iMiclid.  I'his  is  fol- 
lowed by  a  treatise  upon  astronomy  and  mensuration,  also  in  Latin,  in  a 
different  but  contemporary  hand.  The  mensuration  includes  some  w(irk 
on  areas  and  volumes. 

ANONYMOUS.  Latin  MS.,  1384. 

Title.    None.    A  C()mi)utus  manualis. 

Description.  4°,  13.3  X  18.8  cm.,  the  text  being  9  x  13.3  cm. 
33  ff.  (3  blank),  40-44  11.,  written  on  paper. 

The  first  written  folio  has  been  torn,  and  the  opening  lines  are  miss- 
ing. The  work  is  that  particular  kind  of  mediaeval  computus  (see  p.  7) 
in  which  a  finger  mnemonic  system  is  used  (see  p.  34).  It  is  a  copy  of 
an  older  treatise,  the  text  of  which  is  here  written  in  Gothic  characters, 
the  copyist's  notes  appearing  in  a  smaller  hand.  As  in  most  computi, 
the  numerals  are  generally  in  Roman,  both  in  the  text  and  in  the 
comi-nentary,  but  the  date  is  twice  given,  as  follows  (f.  19,  r.)  :  '  ano 
diij  1000.300.80.4';  'anno  dnj  1000.300.80.4'  (see  Fig.  221).  The 
text  is  in  Latin,  except  for  one  page  which  is  in  French.  A  later  owner 
has  written  the  date  of  his  ownership,  1600  (f.  2,  r.). 

ANONYMOUS.  Italian  MS.,  1393. 

Title.  '  CQui  chomincia  ilpologho  d-1  conpoto //ddcorfo  d-1 
folo  3  d-lla  Itmo//    Prolagho.'     (F.  3,  r.,  Fig.  222.) 

Description.  FoL,  22  x  29.7  cm.,  written  in  double  columns, 
each  being  7.5  x  20.5  cm.    70  ff.  (6  blank),  35  11. 

This  is  an  excellent  example  of  a  computus  (see  p.  7).  The  verse 
'  Thirty  days  hath  September,'  the  only  relic  of  the  old  com])uti  now 
familiar  to  most  people,  appears  in  this  manuscript  in  the  following 
Italian  form:    '  Trenta  di   a  noumbre  apile // giugno  &   fettenbre  di 

uentotto  ( )  vno  tutti  glialtrj  fono  trentuno.'   (F.  12,  v.)     (See 

also  p.  33.) 

GIOVANNI,  the  .son  of  Luca  da  Firenze.       Italian  MS.,  1422. 

The  son  of  Maestro  Luca,  a  celebrated  Florentine  arithmetician,  mentioned 
on  p.  468. 

Title.    '  Trattato  di  aritmetica.'     (F.   i,  r.) 

Description.   4°,  14.9  X  21.9  cm.,  the  text  being  9. 3  X  16.3  cm. 

145  ff.  27-30  11. 


444  KARA   ARITHMETICA 


r^^     Jti?  ami  ttfu  ttiSttt-to^^wgitef?^ 
ip<>1  ptr  tti»atttif  m^tu^  nt^nt^  ctjtttu)  |*f 

^STuTBln/t^t  tit  tHflpEie  fintttl^  Vmnti 


Fig.  22  1.    From  the   1384  computus 


MANUSCRirrS 


445 


This  is  one  of  the  best  examples  of  an  early  fifteenth-century  com- 
mercial arithmetic  known.  Florence  was  at  this  lime  an  imjjortant 
financial  center,  and  the  arithmeticians  of  the  city  were  highly  esteemed. 
Several  of  the  applications  found  in  arithmetics  for  the  next  three  cen- 
turies had  their  origin  here.  Subjects  like  equation  of  payments  and 
partnership  involving  time,  customs  like  'days  of  grace'  in  exchange, 
and  forms  like  time  drafts  in  sets  of  two  or  three  can  be  studied  to  good 
advantage  in  the  arithmetical  manuscripts  of  this  period. 


e^« 


v«eoro Jfvfl  tJorjy>  -Tiry^  f^c  fti^  >•« 

%,«7vf>v*<  .fWn9cC«.  -f^TMvyt  9vfto^ 
iU»«  fvrt  .^i^Trn+i?^"^  ore  )<«^ 

^c  ^PwtvoG:  .^n««»<  2'C-^'^  '"♦^ 

yC^   «***«<   S-^^W    f  COp.^^^    ^'^^ 


cPn-  -g^." 


Fig.  2  2  2.    First  pace  oi-  the  1393  computus 


446  KARA   ARITHMETICA 

On  f .  I ,  v.,  the  writer  says  he  proposes  to  treat  of  arithmetic  or  the 
abacus:  ' arifmetricho  volgharemente  e  chiamata  abacho.'  The  word 
'  abacus '  had  come  at  this  time,  in  Italy,  to  mean  simply  arithmetic, 
the  original  meaning  having  been  lost.  Following  a  custom  of  Floren- 
tine arithmeticians,  Giovanni  gives  an  extensive  multiplication  table,  for 
purposes  of  reference,  and  then  begins  at  once  with  examples  in  com- 
pound numbers  and  fractions,  thus  presupposing  a  knowledge  of  the 
fundamental  operations.  These  examples  are  of  a  mercantile  character 
and  constitute  the  entire  portion  devoted  to  arithmetic.  The  examples 
in  the  equation  of  payments  serve,  as  usual,  to  fix  the  date  of  the  manu- 
script. They  all  refer  to  the  years  1418-1426  (ff.  1 13-122).  The 
date  is,  however,  fixed  exactly  by  the  closing  lines  of  a  folio  near  the 
end  :  '  quefto  libro  Ifcriffe  Giouannj  del  maeftro  lucho  dellabacho  e 
finillo  quefto  d  dottobre  1422  — '  (f.  136,  v.).  In  the  first  page  is  the 
date  'ad  28  dottobre  1422.' 

The  book  also  contains  a  section  on  mensuration  and  the  calendar, 
with  curious  illustrations  of  the  months  (PI.  VII).  The  last  five  folios, 
originally  blank,  seem  to  have  been  written  by  a  different  hand  about 
the  same  period. 

ROLLANDUS.  Latin  MS.,  1424. 

A  native  of  Lisbon,  canon  of  Sainte-Ciiapelle,  Paris,  c.  1425. 
Title.    '  Scientia  de  niimero  ac  virtute  numeri.'     (F.  3,  1.) 
Description.    FoL,  21  X  29.8  cm.,  the  text  being  15.5  X  21.5 
cm.     168  ff.,  28-31  11. 

This  is  an  exceedingly  interesting  manuscript,  written  in  the  year 
1424.  It  was  prepared  at  the  command  of  John  of  Lancaster,  Duke  of 
Bedford,  son  of  Henry  IV  of  England,  at  one  time  Protector  of  England 
and  Regent  of  France.  To  him  Rollandus  dedicates  the  treatise  :  '  Illuf- 
triffimo  ac  fereniffimo  principi  metuendiffimo  domino  domino  lohanni 
patruo  domini  noftri  regis  ffrancie  et  anglie  regenti  Regnum  ffrancie. 
duci  bethfordie  Rollandus  fcriptoris  veftre  celfitudinis  phyficus  vlex- 
bonenfis  fe  ipfum  ex  debito  iuramenti.'  In  1423  Lancaster  issued  an 
ordinance  for  the  restoration  of  studies. in  the  University  of  Paris,  and 
it  was  probably  as  a  result  of  this  that  this  textbook  was  written  by  a 
Portuguese  physician,  Rollandus,  who  was  then  a  canon  of  Sainte- 
Chapelle  in  Paris.  The  dedication  sets  forth  Lancaster's  interest  in 
learning  in  France  and  the  status  of  mathematics  at  that  time.  Rol- 
landus covers  all  of  theoretical  arithmetic  as  then  known,  but  takes  up 
no  practical  problems.    He  also  treats  of  irrational  numbers,  a  topic 


r\,  — iojie(jf\c~ 


MANUSCRIPTS  447 

which  is  now  considered  part  of  algebra.  It  is  doubtful  if  there  is  a 
manuscript  extant  which  throws  more  light  upon  the  nature  of  Frenc  h 
university  mathematics  at  the  time  this  was  written.  RoUandus  also 
wrote  a  work  on  surgery  and  one  on  physiognomy.  lie  may  possibly 
be  the  RoUand  who  in  14 10  was  rector  of  the  University.  Since  this 
manuscript  is  evidently  a  copy,  others  must  have  existed,  but  I  have 
found  no  reference  to  them.  Fig.  223  shows 
the  forms  of  the  numerals  used  at  this  time. 

It  should  be  noticed  that  the  numerals  that        ^r (tZ  C2^ii'/, 

have  changed  materially  in  form  since  the  •    ^ 

twelfth  century  are  4,  5,  and  7.    These  are         "  *-^^^)iy< 

shown  opposite  the  letters  l>,  i,  and  /,  re-        €\' •   ^*  ^A  I 

spectively,  in  Fig.  223.     The  changes   in         tl ^7^?^- 

the  other  forms  have  been  more  evidently  U "72^^"$  . 

due  to  the  fashion  in  handwriting.    All  of  J 

the  forms  are,  however,  quite  different  from  O        ~      >  •'/C ' 

the  primitive  ones  found  in  the  cave  inscrip-  W B I  ^2. 

tions  of  India.  y^x S.o^C 

ANONYMOUS.  *t^ ^^^ 

Italian  and  Latin  MSS.,  c.    1430  ^ 61  i 

and  1478.  Q l^^ 

Title.    The  first  folio  is  missing.  ^ 

Description.    Fol,  17  X  22.1  cm.,  the  ^            ^  ' 

text    being    11.2X14.4   cm.     183   ff.,  '^              /^' 
25-28  11. 

This  volume  consists  of  two  Florentine 


manuscripts,  one  on  commercial  arithmetic 


c  —  ;5^ 


and  the  other  on  the  computus.   The  por-  \y i^. 

tion  on   arithmetic   was   probably   written  •»! 2. 

between    1420  and  1444,  these  being   the  .. 

extremes  of  the  dates  in  the  examples  in 

equation  of  payments.    It  is  not  unlike  the         Fk;.  223.    From  tmk 
other  contemporary  Florentine  arithmetics  1424  koli.andus 

described  on  pp.  443  and  464.    The  author 

assumes  the  student's  ability  to  perform  the  fundamental  operations 
with  integers,  although,  after  numerous  multiplication  tables,  he  gives 
(see  Fig.  224)  an  example  under  'Multiplicha  p  modo  de  barichocholo ' 
(the  Florentine  name  for  our  present  method),  and  one  under  'Multi- 
plicha p  modo  de  Quadrato '  (the  'gelosia'  method  of  the  Venetians). 


448 


RARA   ARITHMETICA 


.fc-i.%i 


M'"  I 


,     ^<e —  90  f — 

^'  CO  8i^  ^^^    f  ]J9_ 


'111/0/ 


^9  ^V^">^oc(;. 


Fig.  224.   From  the  anonymous  manuscript  ok  c.  1430 


MANUSCRIP'IS  449 

In  the  latter  part  of  the  work,  as  if  it  were  a  new  topic  ,  the  author  has 
introdiuetl  (f.  126,  v.)  a  chapter  entitled  '  I'artire  a  danda.'  He  chjses 
with  several  folios  on  mensuration. 

The  second  part  (beginning  on  f.  154,  v.)  is  in  a  different  hand.  It 
is  a  computus,  and  from  the  dates  it  would  seem  that  it  was  written  in 
1475.  'Inhere  is  a  third  i)art  (beginning  on  f.  172,  r.)  consisting  of  a  set 
of  religious  verses,  in  a  still  different  hand,  bearing  the  date  1478.  'Ihe 
language  is  Italian,  excepting  for  the  computus  which  is  Latin. 

ANONYMOUS.  Latin  MS.,  c.  1435. 

Title.  '  De  tempore  do  c()mi)()siti()nc  cpiadralis  ct  compositicjnc 
astrolabii  plani,'  etc. 

Description.  Boinid  with  the  'Comptitus  cyronictralis '  of  1476. 
Written  on  paper,  in  a  German  hand.  Some  dates  which  it  con- 
tains indicate  that  the  book  was  written  about  1435.  It  is  related 
to  the  history  of  arithmetic  only  in  the  forms  of  nimicrals  used. 

JOANNES   DE  GMUNDEN.  Latin  MS.,  c.  1439. 

See  p.  1 17. 

Title.    None.    A  treatise  on  the  computus. 

Colophon.  On  f.  17,  v.,  are  the  words,  '  Exphcit  kalendariCi  nii^rl 
Joh'is  gmijnd.' 

Description.  Fol.,  19.  i  X  25.7  cm.,  the  written  part  being  about 
14. 1  X  18.5  cm.  (The  tables  vary  and  are  larger.)  20  ff.,  32-38  11. 
Latin  manuscript.  Written  on  vellum,  in  red  and  black,  about 
1439,  as  shown  by  the  dates  on  f.  16. 

Although  entitled  a  '  Kalendarium,'  it  is  so  much  like  the  mediaeval 
computi  as  to  have  a  place  in  this  list.  It  is  really  a  semiarithmetical 
treatise  on  the  Church  calendar. 

ANONYMOUS.  Latin  MS.,  c.  1441. 

Title.    None.    A  computus. 

Description.  Fol,  21.5  X  28.5  cm.  (varies),  10  ff.  (i  blank); 
the  number  of  lines  to  each  folio  varies  considerably.  Latin  manu- 
script, c.  1 44 1. 

The  first  j)art  of  this  manuscrijit  is  a  computus,  not  very  extended. 
It  relates  rather  to  the  calendar  itself  than  to  the  computations  ui)on 


450  KARA   ARITHMETICA 

which  it  is  founded.    The  last  folio  is  in  a  later  hand,  and  from  the 
tables  which  it  contains  it  seems  to  have  been  written  c.  1524. 

ANONYMOUS.  Latin  MS.,  c.  1442. 

Title.    None.    A  treatise  on  the  planets. 

Description.  Bound  with  the  '  Computus  cyrometralis'  of  1476. 
Written  on  paper,  in  a  German  hand.  It  contains  the  date  1442 
in  two  places  (f.  102).  Its  value  in  connection  with  arithmetic 
is  confined  to  the  forms  of  the  numerals  used. 

JOHANNES  SACROBOSCO.  Latin  MS.,  c.  1442. 

See  p.  31. 

Title.  An  algorismus,  beginning  with  the  words,  '  Omnia  que 
a  primeua  rerum  origine  processerunt.'    (F.  3.  r.) 

Description.  4°,  14.  i  X  20  cm.,  the  text  being  10.5  X  17.6  cm. 
21  ff.,  25-39  11. 

The  first  folio  has  a  picture  of  an  astronomer  with  a  celestial  sphere, 
and  the  name  of  the  student  who  copied  the  MS., '  hainricus  muglinchk  ' 
(f,  I,  r.).  On  the  next  page  (f.  i,  v.)  Mugling's  name  again  appears, 
together  with  three  dates:  'Item  hainric'J  mugling  aftronim9,'  1442, 
1443,  1444.  The  '  Algorismus  '  begins  on  foho  2,  r.  :  '  Omnia  que  a  pri- 
meua//rerii  orign5,'  and  ends  on  f.  10,  v. :  '  Explicat  algorafmus.'  Then 
follows  a  picture  (PL  VIII)  in  colors,  representing  a  master  teaching  his 
pupil  the  Hindu  numerals  from  a  kind  of  large  hornbook,  with  a  motto  : 
'Ich  pin  algorifm9  genant//Das  ...(?)...  hau  ich  in  mein//nes 
hant.'  A  table  explaining  place  value  is  given  on  f .  1 1 ,  r.  Beginning 
on  f.  II,  v.,  is  another  treatise  on  arithmetic,  giving  the  fundamen- 
tal operations  and  some  work  in  progression,  and  ending  with  a  multi- 
plication table  (ff.  18,  V. ;  19,  r.).  A  later  hand  has  added  three  pages 
on  progressions,  rule  of  three,  partnership,  and  interest.    (Ff.  1 9,  v. ;  20.) 

Sacrobosco's  algorismus  was  the  first  arithmetic,  based  on  the  new 
numerals,  written  by  an  English  scholar.  It  consists  of  eleven  chapters, 
viz.  Numeratio,  Additio,  Subtractio,  Mediatio,  Duplatio,  Multiplicatio, 
Divisio,  Progressio,  Perambulum  ad  radicum  extractionem,  Extractio 
radicum  in  cubicis.  For  the  various  editions  of  this  work  see  p.  32- 
The  title  of  the  Paris  edition  of  15  10  is  '  Opvscvlvm  de  praxi  numerorum 
quod  algorifmum  vocant,'  and  the  work  consists  merely  of  four  folios 
containing  the  chapter  '  De  arte  numerandi.'  There  was  also  pubhshed 
at  Antwerp  in   1547   (with   later  editions,  Paris   1550,  Venice   1564, 


MANUSCRIPTS  451 

Wittenberg  1578),  Sacrobosco's  *  Libellvs,  de  aiini  ratione:  sen  vl  vex  a- 
tiir  vulgo,  Computus  ecclesiasticus.'  His  work  on  the  Sjjhere  was  ])iib- 
lished  in  1488  at  Venice.  For  a  discussion  of  the  authenticity  of  tlic 
Algorismus,  see  De  Morgan,  p.  13. 

As  stated  on  p.  31,  the  date  of  the  death  of  Sacrobosco  is  uncertain. 
It  is  either  1244  or  1256,  according  as  we  interpret  certain  lines  on  his 
tomb  : 

'  M.  Christi  bis  C  ciuarto  dcno  {|uater  anno 
De  Sacro  Bosco  discrevit  tempora  ramus, 
Gratia  cui  dederat  nomen  divina  Johannes.' 

MARO  ANTONIO  ROZINO.  Latin  MS.,  1447. 

Title.  '  Qoiies  Marci  Antonij  rozoni  artiu  //  doctoris  sacro  thc- 
ologie  magri  et  papie  phiam  legentis.'     (F.  i,  r.) 

Description.  FoL,  20.3  X  28.1  cm.,  the  text  being  13.7  X  24 
cm.    99  ff.,  32-33  11. 

This  is  not  an  arithmetic,  and  it  has  been  included  in  this  list  only 
because  it  is  semi-mathematical  and  is  bound  with  Bradwardin's  treatise 
on  the  theory  of  proportion.  It  is  a  Latin  treatise  on  the  theory  of  per- 
spective. It  is  written  in  a  clear  Italian  hand,  and  was  part  of  a  volume 
numbered  493  in  the  Boncompagni  sale,  containing  four  manuscripts. 
Some  dealer  has  removed  the  first  of  these  manuscripts,  the  '  Perspectiva 
communis '  of  John  Peckham,  archbishop  of  Canterbury.  A  memo- 
randum in  the  Boncompagni  catalogue,  probably  from  a  leaf  removed 
with  the  first  treatise,  shows  that  the  manuscript  was  copied  in  1447  : 
'  scripte  per  me  antonium  confaronesium  ut  (vocatur?)  de  lavilata  Anno 
diii  M"ccccxlvij.'  The  other  manuscripts  bound  with  this  are  described 
below  and  on  p.  452. 

THOMAS  BRADWARDIN.  Latin  MS.,  i447- 

See  p.  61. 

Title.    None.    A  treatise  on  proportion. 

Colophon.  '  Expliciiit  propor»es  thomi  brardi  //scripte  p  me.' 
(F.  115,  r.) 

Description.  Fol.,  20.3  X  28.1  cm.,  the  text  being  13.7  X  22.8 
cm.     15  ff.,  33  11. 

This  forms  part  of  the  volume  last  mentioned,  and  is  written  in  the 
same  hand.  Bradwardin's  treatise  on  proportion  was  published  in 
Paris,  1495  (p.  61). 


452  KARA   ARITHMETICA 

ANONYMOUS.  Latin  MS.,  1447. 

Title.    A  treatise  on  lenses. 

Description.  FoL,  20.3  X  28.1  cm.,  written  in  two  columns, 
each  being  6.6  X  22.9  cm.    7  ff.,  24-33  H- 

This  rather  early  work  on  optics  forms  part  of  the  volume  mentioned 
on  p.  451,  and  is  written  in  the  same  hand. 

ST.  BERNARD  OF  SIENA.  Latin  MS.,  c.  1450. 

Bom  at  Massa,  Tuscany,  in   1383;   died  at  Aquila  in   1444.     He  was  a 
zealous  founder  of  monasteries,  and  wrote  various  religious  treatises. 

Title.  This  book  of  sermons  begins  as  follows:  '  S"!°  .33.9  .de 
reftitutioe.//  Doiii  .q.  in  quadrageffla  ordo  dicendo:^  p  feptimana 
fequetes  I  li°  de  x'ana  re//ligione  .a.  R'!"  p.  S.  B."^  de  fenif  ordonif 
mino:^  edito.'     (F.  i,  r.) 

Colophon.  '  Explicit  tra//ctatuf  de  ufsif  2  5ctib5  £5115  Bntu5  // 
Bnardini5  (?)  de  fenif.  ordlf  minor:^.'    (F.  129,  r.) 

Description.  4°,  14.8  X  19.8  cm.,  the  text  being  10  X  13.8  cm. 
129  ff.,  part  vellum  and  part  paper.  After  f.  60  the  pages  are 
arranged  in  double  columns,  each  being  5  X  i4-5  cm.,  44-46  11. 

This  beautifully  written  manuscript  of  c.  1450  is  included  in  this 
list  because  it  contains  several  sermons  bearing  upon  the  mercantile 
customs  of  the  time,  including  '  De  usuris,'  '  De  cambiis,'  '  De  con- 
tractibus,'  and  '  De  mercatoribus.' 

Part  of  this  work  was  translated  into  Italian  in  the  fifteenth  century, 
as  appears  from  a  codex  in  the  BibUoteca  Comunale  at  Siena.  One  of 
the  sermons  was  published  by  Riccomanni,  in  the  Scelta  di  Curiosita  Let- 
terarie  inedite  0  rare  (no.  13)  of  Romagnoli,  Bologna,  1862.  This  ser- 
mon, '  Sulle  Soccite  di  Bestiami,'  contains  considerable  information  as 
to  the  business  problems  of  monastic  institutions  of  the  fifteenth  cen- 
tury. A  copy  of  several  of  these  sermons,  made  by  one  Eustachio  da 
Feltre  in  1469,  is  mentioned  on  p.  466. 

ANONYMOUS.  Latin  MS.,  c.  1450- 

Title.    None.     On  the  Quadrivium. 

Description.  4°,  14.7  X  21  cm.,  the  text  being  14.5  X  17  cm., 
47  pp.,  18-40  11. 

This  is  a  Latin  manuscript  on  the  Quadrivium.  It  is  written  on 
paper,  in  a  German  hand  of  about  1450.    It  includes  a  brief  treatment 


MANUSCRIPTS 


453 


of  arithmetic  (5  ff.  +  i  blank,  the  text  occuj)ying  ff.  iS  —  22,  v.),  and  a 
brief  treatment  of  the  calendar  and  the  zodiac  (ff.  15,  r.  —  17,  v.),  'I'he 
geometry  begins  with  modifications  of  Euclid's  definitions  :  '  Pvncf  eft  c^ 
pars  no  eft/I-inea  e  longi^"  fii  latitu"^  2  pfuditate.'  It  closes  :  'CKxplicat 
prim9  liber  euclidis  cu  ometo  capani.' 


JOHANNES  SACROBOSCO.  Latin  MS.,  c.  1450. 

Seep.  31. 

Title.  '  Spera  mundi  .secundum  J()ha//nnem  dc  facrobofco.' 
(F.  I,  r.) 

Colophon.  '  Explicit  tractat'^  dc  fpa  I'm  //  Johem  de  facrobofco 
am.'    (F.  35,  V.) 


Fui.  225.    From  sacrobosco's  Spera.,  c.  1450 

Description.    Fol.,  35  ff.,  written  on  vellum. 

Printed  works  of  this  kind  have  not  been  included  in  tliis  catalogue. 
This  manuscript,  however,  shows  the  state  of  the  numerals  about  1450, 
and  therefore  is  mentioned. 

The  treatise  begins  (f.  i,  r.)  :  '  De  fpera  in  quatuor  capitula  diftin- 
gui9  //  dicentes  p'mo  quid  fit  fpera/quid  eius  //  centrum/  quid  axis  fpere 
/et  quid  fit  po//lus.' 

The  first  four  folios  have  a  marginal  commentary,  closely  written  in 
a  later  hand.    The  figures  are  carefully  drawn  throughout  (see  Fig.  225). 


454  RARA   ARITHMETICA 

There  is  nothing  to  show  the  exact  date  of  the  MS.,  but  the  handwriting 
and  numerals  indicate  the  last  half  of  the  fifteenth  century. 

Sacrobosco's  '  Sphere '  was  published  in  1 488,  and  often  thereafter.  It 
was  the  great  mediaeval  work  on  astronomy. 

ANONYMOUS.  Latin  MS.,  c.  1450. 

Title.    '  Incipiut  floref  Arifmetrice.'     (F.  i,  r.) 
Description.    4°,  15  x  21.8  cm.,  the  text  being  8.4  X  15.5  cm. 
70  ff.  unnumb.  +  2  blank  =  72  ff.,  45  11.     Latin  MS.,  in  a  Ger- 
man hand,  c.  1450. 

This  is  a  theoretical  treatise  on  arithmetic  and  algebra,  written 
respectively  on  the  Boethian  and  Al-Khowarazmian  models.  Only  a 
little  elementary  treatment  of  the  fundamental  operations  (chiefly  mul- 
tiplication) is  given,  the  writer  devoting  most  of  his  attention  in  the 
first  part  of  the  work  to  subjects  like  progressions,  ratios,  and  propor- 
tions. The  latter  part  of  the  book  is  algebraic  and  may  prove  to  be  a 
copy  of  some  mediaeval  work  of  importance.  It  resembles  in  some 
places  the  work  of  al-Khowarazmi,  a  manuscript  of  which  follows  this 
one  in  the  same  volume. 

MOHAMMED  IBN  MUSA  AL-KHOWARAZML 

Latin  MS.,  1456. 
Abu  'Abd  Allah  Mohammed   ibn   Musa  al-Khuwarizml    Born  in 
the  province  of  Khwarazm  (whence  his  name),  died  c.  831.    The  most  cele- 
brated algebraist  of  his  time,  and  the  first  to  write  a  book  bearing  the  title 
Algebra.    Yxom.  his  name  comes  the  word  algoristn  (see  p.  7). 

Title.  '  Liber  mahometi  de  Algebra  et  almuchabila  x  comparif 
et  oppof.'    (See  Fig.  226.) 

Colophon.  There  is  none,  but  f.  numb.  85  bears  the  date  1456, 
and  the  forms  of  the  numerals  and  letters  are  of  that  period. 

Description.  4°,  15  X  21.8  cm.,  the  text  being  12  X  17.2  cm. 
23  ff.  unnumb.  +  i  blank  =  24  ff.,  44-48  11.  Latin  MS.  in  Ger- 
man hand,  1456. 

This  interesting  manuscript  of  the  first  book  bearing  the  name  alge- 
bra is  more  complete  than  the  one  found  by  Libri  in  the  Bibliotheque 
Nationale  (^ffistoire  des  sciences  mathhnatiijues,  I,  note  XII).  It  more 
nearly  resembles  the  one  which  I  found  in  the  Columbia  University 
Library  in  1904,  and  showed  to  be  in  the  handwriting  of  Scheubel. 
The  two  deserve  to  be  edited  and  compared  with  the  Rosen  translation 


MANUSCRIPTS  455 


cf^r^  ,*r»-**^  W^^  -^.A'-^^t  ■»~.«ni^«»».*y»i*^/&V*      / 


ft  Jf^JJZ.*^  ^^^'^  T-**^  or~/i.  ^^,^  ^^«^  i-sAK^r^ 

Fig.  226.    First  page  of  thk   1456  al-khowarazmi 


456  RARA   ARITHMETICA 

(London,  1 831 ).  This  manuscript  is  particularly  valuable  because,  unlike 
the  one  published  by  Libri,  it  has  the  Arabic  numerals  and  the  mediaeval 
algebraic  symbols.  The  et  used  for  plus  so  closely  resembles  the  +  as  to 
leave  little  doubt  that  the  latter  was  derived  from  this  Latin  word.  Like 
Regiomontanus,  the  writer  uses  i9  for  minus. 

This  is  followed  (ff.  numb.  97)  by  a  brief  treatise  on  rhetoric,  and 
by  three  manuscripts  on  mathematics  of  1501,  c.  1475,  and  c.  1550, 
described  on  pp.  480,  468,  486. 

ANONYMOUS.  Italian  MS.,  c.  1450. 

Title.    A  treatise  on  mensuration,  roots,  and  algorism. 

Description.  FoL,  21.6  X  30.3  cm.,  the  text  being  10.5  X  15.8 
cm.  177  pp.  (2  blank),  20-25  11.  Written  on  paper,  in  a  Flor- 
entine hand,  c.  I450- 

This  is  a  general  treatise  on  mathematics,  with  divisions  as  follows : 
A  'praticha  de  Geometria '  (ff.  2-1 1);  fractions  (ff.  13-16);  square 
root  (ff.  17-19,  r.)  ;  cube  root  (ff.  19,  v.,  —21);  ratios  (ff.  22-26); 
algebra  (ff.  26-30);  mensuration  (ff.  31-46);   algorism  (ff.  47-177). 

The  algorism  is  evidently  the  work  of  some  Florentine  teacher,  and 
the  handwriting  is  that  of  the  middle  of  the  fifteenth  century.  It 
includes  the  usual  applications  of  the  period,  such  as  profit  and  loss, 
partnership,  and  interest,  and  it  makes  prominent  the  rule  of  three  and 
the  rule  of  false  position. 

PETER  PAUL  VERGERIUS.  Latin  MS.,  c.  1450- 

Born  at  Capo  d'Istria,  Venice,  c.  1350  ;  died  in  Hungary  c.  1420,  or  pos- 
sibly in  1444.    Bishop  of  Capo  d'Istria. 

Title.  '  De  Ingeniis  Moribus  et  liberalibus  Studiis.  Francis- 
cvs//fenior  auus  tuus  cui9  ut  extant //plurime  res  magnifice  gefte 
ita//et  multa  paffim  ab  eo  fapieter  i// dicta  memorantur.  .  .  .' 
(F.  i,r.) 

Colophon.  '  Petrj  pauli  uergerij  de  uiftinopoli  ad  uber-//tinu5 
Cararienfem  de  ingenijs  moribs  2  //liberalibs  ftudijs  adolei'centie 
liber  felici-//ter.     Explicit.// Amen.'    (F.  23,  v.) 

Description.  4°,  14  X  20.3  cm.,  the  text  being  8.8  X  13.6  cm. 
23  ff.  (10  vellum,  13  paper),  26-36  11.  The  first  10  (vellum) 
folios  are  in  Gothic  script;  the  last  14  are  plainly  written  in 
cursive  characters. 


MANUSCRIPTS  457 

The  treatise  refers  to  '  arifmelicha  '  and  '  geumetria  '  (f.  15,  r.)  anioiif,' 
the  Hberal  arts.    It  is  bound  with  several  other  manuscripts. 

RAYMUNDUS   LULLIUS.  Latin  MS.,  c.  1450. 

R.WMo.M)  l.i;i.i,Y,  Ramon  I.ri.i,.  Born  in  Palnia,  Majorca,  c.  1234;  died 
in  1315.  He  was  a  Catalan  alciieniist,  pliilosojiher,  and  missionary,  and  was 
known  as  '  Doctor  ilhiminatus.' 

Title.  '  Ars  Brevis.'  A  note  by  a  somewhat  later  hand,  written 
on  f .  I,  r.,  reads  :  'Ars  breuis  Raymundi  Lull  quam  scripsit  Pisis 
//in  monasterio  Sancti  Dominici  anno  1307.  a.  c.23.'  The  Ars 
brevis  begins  on  f.  5,  r. :  'Deus  cu  tua  gra  fapla  et  amore  Incipit 
ars  breuis  que  eft  //ymago  artis  ghalis.  Na  ifta  fata  ab  intelectu 
fb'tili  et  fun//dato  ipe  pot  fcire  gnale  arte.' 

Colophon.  'Finiuit  Raymundus  arte  breuen  pifis  in  monaft'io 
fric//ti  dominici  Anno  ab  incarnati(jne  diii.  1307.// E.xplicit  feh- 
citer.'     {¥.  23.) 

Description.  4°,  15.4x20.8  cm.,  the  text  being  9.9  x  13.2 
cm.     24  ff.,  28  11. 

This  is  the  second  part  of  a  manuscript  of  59  folios,  of  which 
the  first  is  the  Sensnale  of  Lullius.  This  and  several  other  man- 
uscripts are  bound  with  the  Vergerius  already  described  (p.  456). 

The  Ars  brevis  was  originally  written  in  1307  ;  this  copy  was  made 
about  1450.  Although  not  an  arithmetic,  the  work  contains  several 
mathematical  definitions. 

JOHANNES  ROS.  Latin  MS.,  1450. 

A  Valencian  priest. 

Title.  '  Artificium  artis  arithmeticac.'  (F.  i,  r.)  '[DJEusqui 
es  unus  .  .  .  InciJD  artifi"V  atis  alfmet'ce  De  alphabeto.'    (F.  24,  r.) 

Colophon.  '  Ad  laudem  //  olpotentis  dei  et  uirginis  marie  .  .  . 
fi//niuit  frat'  Johanis  ros  de  Valencia  puincie  aragonii  hoc  // 
artificiQ  arifmetice  pad  ...  I  loco  fratrum  mino^  de  ofpitali  // 
1450  die  .5.  Januarij  I  uigilia  epiphanie  domini.'     (F.  14  =  37,  r.) 

Description.  4°,  15.1  x  19  cm.,  the  text  being  9.2  X  13.7  cm. 
36  ff.,  28  11.  Bound  with  the  Lullius  and  Vergerius  described 
above  and  on  p.  456. 


458  RARA   ARITHMETICA 

This  rather  early  Spanish  monastic  treatise  on  algorism  is  the  third 
of  three  manuscripts  in  the  same  hand,  written  c.  1450,  of  which  the 
first  two  are  mentioned  on  pp.  456,  457.  It  begins  on  f.  24,  r.,  and 
ends  on  f.  37,  r.  After  some  definitions,  tables,  and  computus  figures, 
the  author  takes  up  the  '  nine  subjects '  :  '  None  f'biecta  ponut  i  arif- 
metica '  (f.  3  =  26,  v.).  These  are  treated  very  briefly.  This  is  followed 
by  a  religious  work,  in  the  same  hand  (to  f.  57). 

ANONYMOUS.  Latin  MS.,  1442. 

Title.    '  De  ymagine  mundi.'     (F.  2,  r.) 

Colophon.  '  Cdeo  grat.  aiij.  (amen  })  Aiio.  dnj.  1442  .  .  .  CEx- 
plicit  lib. 9  de  ymagine  mundi  deo  grat  .  .  .' 

Description.  8°,  14.5  X  21.5  cm.,  the  text  being  1 1.5  x  19  cm. 
158  ff.  in  the  entire  manuscript,  1 1  ff.  in  this  portion.  The  other 
portions  are  described  on  pp.  477,  478. 

This  is  the  first  manuscript  in  a  collection  of  156  folios,  and  begins 
with  the  lines  '  I.  H.  S.//Cde  ymagine  Mundi '  (f.  2,  r.).  It  is  written 
in  Latin  and  treats  of  physical  and  descriptive  geography.  It  has  some 
interest  in  the  history  of  arithmetic  through  the  Roman  numerals  which 
are  generally  employed  except  in  the  case  of  large  numbers.  This  manu- 
script and  the  others  bound  with  it  are  described  in  Narducci's  catalogue 
of  the  Boncompagni  manuscripts  (Rome,  1862,  no.  81 ).  As  there  stated, 
at  one  time  it  belonged  to  Alessandro  Padovani,  a  celebrated  collector 
of  the  sixteenth  century. 

ANONYMOUS.  Italian  MS.,  c.  1456. 

Title.    The  title  is  missing. 

Description.  4°,  14.6  x  21  cm.,  the  text  being  11  x  15.6  cm. 
Ill  ff.,  (12  with  drawings),  30  11.  No.  168  in  the  Boncompagni 
sale  catalogue. 

This  is  a  business  arithmetic,  written  in  northern  Italy,  and  com- 
pleted, as  appears  from  a  note,  July  15,  1456.  As  in  most  of  the  busi- 
ness arithmetics  of  the  period,  the  first  pages  contain  a  set  of  tables, 
these  being  followed  by  a  discussion  of  the  fundamental  operations  with 
denominate  numbers.  Although  written  in  1456  it  is  probably  a  copy 
of  an  earlier  work  of  about  1420,  for  the  examples  in  the  equation  of 
payments  involve  dates  from  1418  to  1425  (ff.  78-84).  The  work  is 
also  interesting  because  it  contains  the  early  form  of  the  sign  %  (see 
Fig.  227)  already  mentioned  on  p.  439.    The  column  tables  used  by 


MANUSCRIPTS  459 

the 

^2. 


merchants  in  their  multipHcation  '  per  colonna,'  and  common   in  the 
Italian  manuscripts  of  this  nature,  are  shown  in  Fig.  228. 


^ 


Fk;.  227.    From  an  Italian  arithmetic  ok  c.  1456 
ANONYMOUS.  Italian  MS.,  c.  14C0. 

Possibly  by  Raffaele  Canacci,  a  Florentine  mathematician. 

Title.  There  is  none  given,  but  the  work  is  a  general  treatise 
on  mathematics. 

Description.  FoL,  28.2  x  39  cm.,  the  text  being  16.5  X  28  cm. 
2  blank  +  322  numbered  ff.  =  324  ff.,  51  11.  Italian  manuscript, 
c.  1460. 

This  is  an  Italian  manuscript,  beautifully  written  on  vellum,  with 
finely  executed  initials  in  colors  and  gold  at  the  beginning  of  each  of 
its  sixteen  books.  It  belonged  at  one  time  to  Libri,  and  later  to  Bon- 
compagni.  Narducci  describes  it  in  the  catalogue  of  the  latter's  man- 
uscripts (no.  14).  The  author  begins  (f.  i)  with  a  description  of  the 
work  :  '  Come  e  in  che  modo  eldetto  trattato  e  diuifo/  cioe  cio  che 
lopa  cotiene.'    The  successive  chapters  are  as  follows : 


460 


RARA   ARITHMETICA 


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Fir 

.  228. 

From  an  Italian  arithmetic  op' 

c.  1456 

MANUSCRIPTS 


461 


I.  '  Qui  chomincia  el  pimo  libro  del  detto  trattato,  &  pima  ]K)ne  la 
diuifione  del  detto  primo  libro/ lacjuale  e  achapitoli/cioe  c  diuifo  i  .4. 
capitolj.'    (F.  I,  r.)    The  four  '  capitoli '  are  as  follows: 

1.  'El  primo  capitolo  del  pimo  libro/ done  fimoftra  lordiiie  e  modo 
del  numerare  le  fighure  chelufano  afcriuere  enumeri.'  (F.  i,  v.)  In  this 
are  explained  the  Hindu-Arabic  notation,  the  nine  '  figure  fignifichatiue,' 
and  the  0,  '  che  i  arabia  fidice  gero.' 

2.  The  title  of  chapter  2  is  wanting,  f.  3  having  for  some  reason  been 
left  blank.   It  related  to  the  addition  of  integers  and  compound  numbers, 

3.  'Qui  chomincia  el  terz(;o  capitolo  del  pimo  libro/ done  fitratta 
del  modo  &  hordine  del  trarre  el  numero  minore  del  numero  maggiore.' 
(F.  5 ,  r.)  The  method  is  that 
of  borrowing  and  repaying. 

4.  'El  quarto  capitolo 
del  primo  libro  di  quefto 
trattato/  Doue  fitratta  del 
modo  &  hordine  del  0chare 
e  numerj.'  (F.  6,  v.)  The 
column  form  of  the  multipli- 
cation table  is  first  given, 
and  is  followed  by  various 
methods  of  multiplying.  The 
names  '  El  berichuocholo  '  (f. 
10,  V.)  and  'p  quadrato ' 
(f .  II,  r.)  show  the  work  to 
be  Florentine  rather  than 
Venetian,  and  the  handwrit- 
ing and  the  numerous  refer- 
ences to  Florence  confirm 
this  fact.  The  method  '  per  quadrato  '  is  shown  in  Fig.  229.  Five  pages 
of  multiplication  tables  are  given,  such  an  elaborate  treatment  having 
been  rather  common  in  the  arithmetics  of  that  city.  Division  is  not 
treated  in  Book  I. 

II.  'El  fecondo  libro  del  detto  trattato.  Nelquale  fi  conticne  la 
natura  &  proprieta  De  numerj.  &  prima  come  e  diuifo  ^:  aprj  lontel- 
letto.'    (F.  17,  r.)    This  is  divided  into  two  chapters: 

1.  'El  pimo  capitolo  del  fjo  libro/  Nelqual  fitratta  la  natura  che  a 
ife  el  nuo  con  diffinitionj  fapute.'  (F.  17,  r.)  This  relates  to  such  ancient 
classifications  of  number  as  odd  and  even,  prime  and  composite. 

2.  '  El  fecondo  capitolo  del  f>o  libro/ Doue  fitratta  de  numerj  nomi- 
nati  per  nomj  apropiati  alle  fighure  geometre.'    (Figurate  numbers.) 


229.    Multiplication  'per 
quadrato,'  c.  1460 


462 


KARA   ARITHMETICA 


III.  'El  terzgo  libro  della  praticha  darifmetricha.  Nelquale  fitratta  el 
modo  di  partire  p  nuj.  &  pima  1  che  forma  e  modo  el  detto  libro  e  diuifo.' 
(F.  23,  r.)    The  subject  of  division  is  now  taken  up  in  three  chapters  : 

I.  By  the  use  of  the  table, '  el  modo  di  partire  per  gli  numerj  fcripti 
i  fulle  librettinj.'    (F.  23,  v.) 

2.  The  '  ripieghi '  method  (f.  28,  v.),  by  the 
successive  factors  of  the  divisor. 

3.  Long  division.  (F.  32,  r.)  The  author 
gives  not  only  the  galley  method,  but  an  early 
description  of  the  method  '  a  danda,'  substan- 
tially our  present  plan  (see  Fig.  230). 

IV.  This  treats  of  common  fractions.    (F. 

V.  Ratios,  '  quantita  proportional].'  (F. 
46,  V.) 

VI.  Mercantile  arithmetic  (f.  58,  v.),  with 
numerous  references  to  the  customs  of  Floren- 
tine merchants.  Sixty-six  large  folios  (132  pp.) 
are  given  to  this  book. 

VII.  The  rule  of  false  position  :  '  El  feptimo 
libro  diquefto  trattato  nelquale  fitratta  del 
modo  delafoluere  de  chafi  p  lofemplice  modo 
delchatain  che  p  moltj  fidicono  principij  del 
chatain.'    (F.  124,  v.) 

VIII.  Simple  and  compound  interest  (f. 
134,  v.),  with  2  pages  (4  incomplete)  of  tables 
giving  the  interest  on  ^100  for  1-21  yrs.  at 

rates  varying  from  '  5  p  100.  laiio '  (f.  140,  v.)  to  '  40  p  100.  lafio  '  (f. 
150,  v.).  Equation  of  payments  is  also  presented  in  this  book  (f.  152, 
v.),  the  dates  ranging  from  1458  to  1464. 

IX.  False  position  as  treated  by  Leonardo  of  Pisa.  (F.  170,  v.) 
Leonardo  is  mentioned  in  PI.  IX. 

X.  Miscellaneous  problems.  (F.  176,  r.)  These  are  largely  tradi- 
tional examples,  and  include  the  hare  and  hound,  finding  numbers  satis- 
fying given  conditions,  problems  about  eating,  the  jealous  husbands,  and 
the  testament  complication. 

XI.  Proportion,  based  on  Euclid  V.    (F.  225,  r.) 

XII.  Algebra, '  della  regola  della  algebra.'  (F.  233,  v.)  Unfortunately 
few  of  the  figures  for  this  book  were  drawn.  The  treatment  is  rhetor- 
ical, practically  no  symbolism  being  used. 


Fig.   230. 


From    an 
italian  manuscript 

OF    C.    1460 


CCUx^v 


^^ 


1      "' 


anui/c  ui«U|[»vnxt  <^\*A^i  cornc  vL«5u.nto  rvclla  l\T,*-a..' 
.  0-bi*in»E  (Vruj  ^tt«  -nui  (\«»T\pUce/'?po  cattle  (IcopoTigoTui  ftilamSVc  Vv 


LL . -^ ,       ^.-        ^        , 

^\          TiA  atCo  riflertto  «iUiCC»^/ Tveall*  «ti(»  c-uoe  «>>>«  K-V-nvio  ftofc)  papuK  U« 

Wl  lo-nvCo  (Vr^pliet  ctx>«>>wi»fr»  «:a»«H»t«  c*>«rtpaToU  ^l*<»»  i>«\,T>r.*o  ^ 

6>«IC»TV<^  (^LLOftUoTlO  f»p«C-  U^TltWb.    C^vU^^- 

»t\o  loc!ui«V«  <lLai«iali«Tn>rnVo  \  C  woV  (\pv\P  fa 


a 


mbotMAtJa.eio'r  rop> 


vnai'bx'tK 


f»m^Uo-  O^Vu«l¥*j-\   ct>tipc(Vi    Otv^^ 


Platk  IX.    From  an  anonymous  manlsckif 


1460 


MANUSCRITTS  463 

XIII.  Algebra  continued,  '  la  regola  dc  Algebra  amiuabale.'  ( K.  279, 
r.)  This  is  a  very  interesting  treatment  of  the  subject '  fecondo  ghugliel- 
mo  de  lunis'  and  '  Lionardo  pifano,'  and  it  throws  some  light  upon  alge- 
bra as  studied  in  the  fifteenth  century.     (See  Plate  IX.) 

XIV.  Algebra  continued  (f.  295,  v.),  according  to  Master  Hiagio 
(' certj  cafi  che  ferine  m?  biagio  nel  fuo  trattato  di  i^ratica')  of  1340, 
Master  Gratia  de  Castellani  ('  fecodo  che  ferine  m.'  gratia  de  caftellanij, 
and  Leonardo  of  Pisa. 

XV.  Algebra  continued  (f.  312,  r.),  according  to  certain  *  maeftrj 
antichi,' viz. :  '  Maeftro  paolo,'  '  m?  Antonio,'  '  m'.'  giouaiij,'  '  leonardo 
pifano,'  'm?  biagio  che  circha  al.  1340.  anj  morj,'  '  m'.'  paolo  fiorj  che 
circha  al.  1360.  duro,'  '  m'.'  michele  padre  di  m°  mariano,'  '  m"  lucha,' 
and  '  un  altro  m'.'  biagio.' 

XVI.  This  is  missing,  the  manuscript  ending  with  Book  XV  not 
quite  completed.  On  f.  i,  r.,  there  is  mentioned  the  '  fedecimo  e  ultimo' 
chapter,  and  this  might  have  contained  the  name  of  the  author  had  the 
work  been  finished. 

EUCLID.  Latin  MS.,  c.  1460. 

See  p.  II. 

Title.    None.    The  first  book  of  the  Elements. 

ColopJion.  '  CExplicat  prim'»  liber  euclidis  cii  omcto  capani.' 
{F.  14,  V.) 

Description.  4°,  14.7  X  2  1.2  cm.,  the  text  being  9.9  x  i  5.8  cm. 
22  ff.,  18-40  11. 

This  manuscript  of  the  first  book  of  Euclid,  with  the  commentary  by 
Campanus,  has  been  included  on  account  of  the  forms  of  the  numer- 
als used.  It  is  written  in  a  German  hand  of  c.  1460,  and  is  bound 
(fif.  1-14)  with  the  two  manuscripts  next  described. 

ANONYMOUS.  Latin  MS.,  c.  1460. 

Title.    None.     A  treatise  on  the  calendar. 

Description.  See  the  preceding  manuscript.  This  is  ff.  15-17 
of  the  volume 

This  manuscript,  bound  with  the  Euclid  just  described,  was  probably 
written  by  the  same  hand.  It  is  a  brief  treatise  on  the  calendar,  and 
was  intended,  as  usual,  for  the  Church  schools.  In  it  occurs  the  date 
1460. 


464 


RARA   ARITHMETICA 

Latin  MS.,  c.  1460. 


ANONYMOUS. 

Title.    None.     On  the  Quadrivium. 

Descriptio7i.    See  p.  463.    This  is  ff.  18-22  of  the  volume. 
This  is  a  general  treatise  on  the  quadrivium,  and  therefore  contains 
a  chapter  on  arithmetic, 

BENEDETTO  DA  FIRENZE.  Italian  MS.,  c.  1460. 

A  Florentine  arithmetician  of  the  first  half  of  the  fifteenth  century. 

Title.  '  Inchomincia  el  trattato  darifme//tricha  efpelialmete 
quella  pte  //  che  e  fotto  pofta  alia  mercatatia  //  e  comminciando 
alnome  didio.'     (F.  11,  r.) 


>  -t  1 

\'^0t 

^C^ 

^<fC 

XX «; 

t        1 

^     1 

1     -x. 

-I.C(? 

-C89 

-h-^ 

^C^ 

400 

t       6 
1      0 

1      -7 
,      7 

1      8 

1     9 

X.    0 

4^.1 

<?.8<V 

c;x9 

'i-i^ 

^x5 

-1-    t 

-I.     -L 

C-j€ 

-7-^9 

789^ 

8<H 

900 

-c     C 

T.     C 

-%.    8 
-1-  8 

-u    9 

a,   0 

o,6y 

lev^ 

to8f 

u<iC 

ix-^ 

^     ' 

Tables  from  benedetto  da  firenze 


Description.  8°,  8.3  X  11.7  cm.,  the  text  measurement  vary- 
ing. 2  ff.  missing  +  348  ff.  unnumb.  =  350  ff.,  20 11.  Manuscript 
on  parchment. 

This  is  a  parchment  codex  written  about  the  middle  of  the  fifteenth 
century.  It  lacks  the  first  two  folios  and  possibly  also  a  few  at  the  end. 
It  is  one  of  the  best  examples  of  the  mercantile  arithmetics  in  Italy 


MANUSCRIl' rs  465 

preceding  the  printed  works.  The  author  Ijcgins,  as  usual,  with  several 
pages  of  tables  (see  Fig.  231),  the  multiplication  table  including  the 
prime  numbers  below  fifty.  There  is  also  a  table  of  scjuares  and  one  for 
the  multiplication  of  compound  numbers.  Benedetto  presupposes  that 
the  reader  is  able  to  perform  the  fundamental  operations  with  integers, 
and  he  begins  at  once  with  operations  on  compound  numbers  and  frac- 
tions.   The  a])plications  include  exchange,  partnership,  and  ecinatioii  of 


«(iUiW^V>>Axa*;U7>i'n«»«tt  y*bu.(i>t?.i-iuj«^M80o  "bile 


Fig.  232.    From  Benedetto  da  firenze 

payments,  the  dates  in  the  problems  including  the  years  from  1460  to 
1464.  The  latter  part  of  the  book  contains  a  number  of  such  traditional 
problems  as  those  of  the  grains  of  wheat  on  a  chess-board,  the  couriers, 
the  hare  and  hound  (see  Fig.  232),  the  jealous  husbands,  and  the  testa- 
ment of  the  dying  man.  The  author  closes  with  a  brief  treatment  of 
mensuration. 

ALBERT  OF   SAXONY.  Latin  MS..  1462. 

See  p.  9. 

7/V/r.  '  Tractatus  proportionum.  Incii)iut  pportioef  copo^^  a 
dno  albertutio.//  [  ]  Roportio  coitr  da  e  duo^  ;Dperito^  in  a^"  t'rio 
unico//ad  imiice  hitudo.'     (F.  i,  v.) 

Colophon.  '  Explicat  pportioef  ppo'*^ // p  reucdo  nirt)  alberto 
defifona  //  Finis  //  '  (F.   12,  v.) 


466  RARA   ARITHMETICA 

Description.  4°,  14.5  X  21.8  cm.,  the  text  being  11.8  X  13.8 
cm.  (varies).  19  ff.,  23  11.  Bound  with  the  Vergerius  and  Lullius 
mentioned  on  pp.  456,  457. 

This  is  the  treatise  on  proportion  that  was  printed  without  date,  prob- 
ably at  Venice  c.  1478  (p.  9).  It  is  followed  by  another  treatise  in  the 
same  hand,  'De  latitudinibus  formarum,'  which  bears  the  date  1462. 

ST.   BERNARD  OF  SIENA.  Latin  MS.,  1469. 

See  p.  452. 

Title.    None.    Sermons  of  St.  Bernard  (Bernardinus)  of  Siena. 

Description.  4°,  15  X  20.6  cm.,  the  text  being  1 1.5  X  14  cm. 
130  ff.  (12  blank),  28  11. 

Editions.  The  works  of  St.  Bernard  were  first  printed  in 
Venice  in  1591,  4°,  and  again  in  Paris  in  1636,  2  vol.,  fol.  Some 
of  his  sermons  were,  however,  printed  in  Florence,  and  some  in 
Venice,  in  1495. 

This  manuscript,  containing  several  of  the  sermons  of  St.  Bernard, 
was  copied  by  one  Eustachio  de  Feltre  in  1469.  Two  dates  are  given 
showing  the  completion  of  parts  of  the  work,  September  22,  1469,  and 
October  6,  1469.  For  the  reasons  for  including  this  manuscript,  see 
p.  452. 

ANONYMOUS.  Italian  MS.,  1473. 

Title.    '  Trattato  della  Arithmetica.'     (F.  i,  r.) 

Description.  Fol.,  16.7  X  23.7  cm.,  the  text  being  10  X  16.2 
cm.     190  ff.,  35  11. 

This  treatise  was  composed,  as  the  first  folio  states  (P'ig.  233),  in  the 
year  1473.  I^^  the  examples  in  partial  payments  the  dates  given  are 
about  1490,  and  this  particular  manuscript  may  have  been  copied  about 
that  time.  There  is,  however,  one  example  with  the  date  1392,  which 
would  seem  to  indicate  that  at  least  part  of  the  book  was  copied  from 
some  earlier  writer.  There  is  also  (f.  183)  a  brief  treatment  of  the 
calendar  with  two  dates  1443.  On  f.  180  there  are  also  the  dates  1452 
and  1453.  In  the  margins  some  sixteenth-century  dates  have  been  added 
by  a  later  owner. 

The  work  is  beautifully  written  on  vellum.  The  first  part  consists 
entirely  of  arithmetic,  treated  from  the  mercantile  standpoint.  That  it 
is  a  copy  of  an  earlier  work  also  appears  from  the  fact  that  a  few  pages 


MANUSCRIPTS  467 


C /LAverruD  ovinctpio  AlnOWC  c-Ii  dio  "^clc  wfAnijxiniX  (lie 

nt<iAva  AX  $CA  MM'i\  '^dai  bcxtfo  m^  ic^ Gioiuixmi 

. bci^ilh.  '^c-lelo  axX)oihAo  o^clcl  aaan(7o.lifh\-'^clcl  q\o 

i-iQ(?XiTno  iondpc_ddi  ct^oihoU  A^i^Scb  pu'.ti-o 

Sfo  CvcfoLntio  AtxCxyio    SAxtino  <=')f\xttovio  <=\dal 

poxA  vncxa.  «jyiA  h  uttxtCL  ?  i(i^^rrJ>fxmX.  i-CJuiierLfxX  di  KtHa  I^^ 

J310  ^nciipnn'pio  mc^o  <^fTTia  acDnoidi-no  ai-dtix  Jthinae- 
titWx  c|itjJ!i  n-iodt  cb  fitno  dllovo  UtttlaQ^ve.tte.renj/\  $Hn+i)a"' 


QytcTo  /?A  co/Jji! -ftittL  Iuntrji;i li  R^Komlt  liijrjifi 


_tD  cUIa.  ncifiii-xJiiiolcve.  ifhi built:  uLovo'\)i 


iitimc 


rlt+Jtuii  ^eza'HA_xliiic2yLjTi'i(| ita.lt  .illino  hcz,e 


M-nti 


rvecMi  V)iii.'/rA  •fill.lCJi  nicnhi  dmturio  Uiiofit-mjofcv  di  cmdlj 
e.- piiv  clj(e.(7niuv(z.cjuGlLo  cF/i^  loi'o  d^fo  dA  Un<^'K\.iv\    cociala 
WiU  wlormiA  iootuTviiA_-c(Wc?iidL.  (cmpic  iia?ia_j<ujii(ulx  ^^\ 
cLa_dG.viofpcim\Ynenhi.  c>i  fpivi  (omio  cIcLbiUTmijmo  ciypohi.-z.tf- 


¥iG.  233.    First  page  of  the   1473   Tmttato 


468  KARA   ARITHMETICA 

are  left  blank  for  subsequent  insertions,  the  scribe  having  written  :  '  Hie 
dificiunt  quatuo // chartse  q  non  funt//i  exemplario  '  (f.  65,  v.)  The 
second  part  of  the  work  consists  of  practical  geometry  and  mensuration 
(ff.  103  to  128).  The  third  part  (f.  129)  relates  to  irrational  numbers. 
On  f.  135  begins  the  '  Regole  de  la  Arcibra,'  a  chapter  on  algebra. 
There  is  nothing  to  indicate  the  name  of  the  author  or  the  copyist. 

ANONYMOUS.  Latin  MS.,  c.  1475. 

Title.    'Be  duplirj  Arti  Viforir.'    (F.  i,  r.) 

Description.  4°,  15  X  21.8  cm.,  the  text  being  10  X  17  cm. 
(varies).  1 1  ff.  +  i  blank  =12  ff.,  40-45  11.  Written  in  a  Ger- 
man hand  of  c.  1450-1475.  Bound  with  the  manuscript  of 
Mohammed  ibn  Musa  (p.  454). 

This  manuscript  gives  some  account  of  gauging,  and  closes  with  a 
brief  treatment  of  trigonometry.  The  medifeval  numeral  forms  are  used 
throughout.  A  copy  also  appears  in  the  1501  manuscript  described  on 
p.  480. 

LUCA  DA  FIRENZE.  Italian  MS.,  c.  1475. 

Maestro  Luca  da  Firenze  lived  in  the  fifteenth  century,  and  was  the  son 
of  the  celebrated  Florentine  arithmetician  Matteo,  who  was  born  in  the 
fourteenth  century. 

Title.  '  Inprencipio  darte  dabaco.'  (F.  i,  r.)  F.  2  begins: 
*  Inprencipio  darte  dabacho  fecondo  loftile  dinfegniare  del  ma-// 
eftro  luca  di  Matteo  da  fiflrenge.'  (Fig.  234.) 

Description.  4°,  16.8  X  23.3  cm.,  the  text  being  11.7  X  13.8 
cm.    46  ff.,  29  11. 

Although  Fabbroni's  Sforia  dcIP  Universita  di  Pisa  (I,  97)  says  that 
Luca's  son  Giovanni  went  with  Lorenzo  dei  Medici  to  the  University 
of  Pisa  in  15 15,  thus  putting  his  birth  about  1495  and  Luca's  birth 
perhaps  about  1450,  I  feel  that  either  this  is  incorrect,  or  it  is  to  some 
other  Giovanni  and  Luca  that  he  refers.  One  of  the  best  evidences  of 
the  date  of  an  arithmetic  is  found  in  the  dates  given  in  its  problems. 
Authors  usually  mention  years  that  are  not  remote  from  the  time  when 
they  write,  and  in  the  examples  in  equation  of  payments  (f.  29,  v.)  Luca 
uses  dates  from  1410  to  1441.  I  therefore  think  that  either  he  copied 
a  problem  from  his  father  (Matteo),  or,  what  is  more  probable,  he  him- 
self wrote  about  that  time,  say  c.  1425,  and  that  in  either  case  this  Gio- 
vanni was  a  descendant  but  not  a  son. 


MANUSCRIPTS 


469 


The  arithmetic  resembles  numerous  others  written  in  Florence  about 
this  time,  such  as  those  described  on  pp.  443  and  464.  The  author 
begins  with  the  fumlamental  operations  and  follows  thc-se  by  a  treatment 


•trxotfa  roflt  crocz.  'rrwtlh-a'tAen-zJc,  Jivii<iat.a  ZCTcji/Mg-nicto.  p>tfr- 

tnnno-tt    dliWj  2.^'^^'^ ''»*''3*'^'^  "'^^'■'^■"^  2  Pclja^  qiie-fHi  r+jofi- 
JVmocjUo -no   -pn-ic.  cDO'""pgv»ic.  ■Jx.nt+fZa.Tno  p-ziTntv  doUcp - 


•in.  .\_ 


^      -7 


)a^\a'mo  l^patA    Jrja.iU^'  avitntv  ^^ww  ttat-' 
rrvo  tvioga  <vrr»<M-io  '»nevrtcl)Fu  prttvicndo   i-jvm*  — 

fticB-ntitvprh,  •maAa.^rmo/  ^aprv\^Ayn^  ^cv '\-nirt, 


f^T<vio*\icv»no  ftMpexA  jvntB/nda-ia  faJirfa  -p^yrra.  rorifoTo  ^^iSjv 
•fa- •noV'Tw.hflticifllno  Bopa  pa-lo-tdvnc  vnfVnc  nK«.(Ji.Jte  ^icoio 
f^ana.  ctjcUw  l>'2A7r<v  ^M-^n-  ttof>n/  <ifvmcvn«<i1•^/^^^^    evndciTiio  M<n-(i 


Fig. 


:34- 


I'CA    DA    FH^ENZF, 


of  fractions  and  denominate  numbers,  closing  with  a  series  of  applica- 
tions to  the  business  problems  of  the  time.  Subjects  like  ecpiation 
of  payments,  partnership,  and  exchange  are  given  the  most  attention. 
The  handwriting  indicates  that  the  manusc-ript  is  a  copy  made  about 


470  RARA   ARITHMETICA 

1 47 5-1 500,  The  symbols  in  Fig.  235  may  also  throw  some  light  on  the 
disputed  origin  of  our  symbol  for  dollars,  $.  No  one  seriously  considers 
such  fanciful  theories  as  the  combination  of  U  and  S,  or  the  Spanish 
banner  about  the  Pillars  of  Hercules.  The  symbol  first  appears  in  print 
in  T/ie  American  Accomptant,  by  Chauncey  Lee  (Lansingburgh,  ijg"]), 
but  in  a  very  different  form  from  that  now  used.  It  was  common 
among  merchants  for  some  time  before  it  was  cast  in  type  form,  for 
a  note  in  one  of  the  early  American  arithmetics  says  that  the  symbol 
was  in  use,  but  that  there  was  no  type  for  it.  The  third  edition  of 
Pike's  arithmetic  (1798)  uses  m.,  c,  d.,  D.,  and  E.,  for  mills,  cents, 
dimes,  dollars,  and  eagles,  but  DaboU's  School )nafter''s  Affiftant  (4th  ed., 
1799,  p.  20)  gives  the  symbol  %  very  nearly  in  its  present  form.  Now 
whence  come  symbols  like  this  ?  If  they  are  invented  de  novo  it  is 
usually  easy  to  find  their  first  appearance,  as  in  the  case  of  symbols  like 
TT,  c,  and  /  (for  ■\J  —  \).  But  mercantile  symbols  usually  develop  slowly, 
like  ^  from  libra,  /-  from  the  old  s  (/)  from  soldi  or  solidi,  and  d  from 
denarii.  So  it  is  probable  that  $  was  simply  developed  from  some 
earlier  symbol  of  value,  such  as  that  for  pounds  or  reales.  Now  the 
symbol  for  pounds  (libra,  lire)  has  various  forms,  appearing  in  England 
as  £  or  lb.,  but  generally  in  Italy  as  L,  or  lb.  with  two  bars  across.  The 
former  is  seen  in  Fig.  243,  from  a  manuscript  of  1545,  and  the  latter  in 
Fig.  216  from  Dagomari's  work.  The  latter  form,  in  the  fifteenth  and 
sixteenth  century  manuscripts,  appears  as  practically  our  dollar  sign,  as 
shown  in  Figs.  235,  237,  246,  and  it  is  not  improbable  that  our  early 
American  merchants  used  it  for  the  new  unit  of  value,  the  dollar,  just  as 
the  /-  for  shilling  is  still  used  in  many  parts  of  our  country  for  12^  c, 
although  the  original  meaning  is  entirely  lost.  The  symbol  came  into 
general  use  in  printed  books  between  1800  and  1825. 

ANONYMOUS.  Latin  MS.,  1476. 

Title.    '  Computus  cyrometralis.' 

Description.  4°,  15.5  X  20.9  cm.  A  volume  of  180  ff.,  con- 
sisting of  four  treatises,  the  other  three  being  elsewhere  described. 

The  first  of  the  four  manuscripts  is  a  computus  of  the  usual  fifteenth- 
century  type.  (See  Fig.  236.)  In  it  the  Latin  original  of  the  verses 
beginning  '  Thirty  days  hath  September  '  appears  (f.  6)  as  follows  : 

'  Ja.  mar 
ma.  iul.  aug.  oc.  de.  deca- 
trib9.  et.  vno.  alij.  trigeta. 
f3.  februus.  octo.  viginte.' 


MANUSCRIPTS 


471 


< 


'"" 

n 
17 
17 

13 

Xfo 

■" 

18 

«9 

T,0 

^ 


17 
n 
n 

« 

->vi^ 

18 
19 

^1 

'9 

^I 

^l^i'o 

-v-o 

4M 

.^^It 

Fig. 


!35- 


"70  ^"^4: 

_.-7?*^-^l 


Tahles  from  luca  da  kikenze 


472 


RARA   ARITHMETICA 


Fic.   236.    From  the   1476    Computus  cyromctralis 


MANUSCRIPTS  473 

(Sec  also  p.  :^;i.)  It  (iocs  not  start  here,  however,  l)ut  goes  Ixick  at 
least  as  far  as  Sacrobosco  (13th  cent.,  see  ]>.  31),  in  whose  Coniinitus 
(see  p.  45  I )  it  appears  as  follows  : 

'Sep.  No.  lun.  Ap.  triginta  dato,  reliquis  ma,i;is  iino. 
Ni  fit  Biffextus,  Februus  minor  efto  duobus.' 

(1545  edn.,  f.  K  6.) 

JACOBUS  OBERNHEYM.  Latin  MS.,  1476. 

A  Niirnberg  computist  of  c.  1431. 

yVV/r.    '  Computus  norembergensis.' 
Colophon.    '  Anno  dni  1476  completo.' 

Description.  See  the  preceding  manuscrij)!,  with  which  it  is 
bound. 

This  is  a  German  computus  written  by  the  same  hand  as  the  preced- 
ing. It  is  evidently  a  copy  of  a  work  written  in  143 1,  for  the  following 
statement  appears  on  f.  65  :  'Anno  diii  m°cccc"xxxi"  quarta  feria  p9 
judica  finit<)  e  liber  ifte  per  me  Jacobu  Obernheym.' 

ANONYMOUS.  Latin  MS.,  1477. 

Title.  None.  A  calendarium.  The  first  folio  begins  'Albeto 
belo  caro  decagolo  ca''  nobilis.' 

Deseription.  4°,  15.4  X  20.4  cm.,  the  text  being  10.5  x  16.4 
cm.  14  ff.,  17  11.  Bound  with  the  Vergerius  and  Lullius  de- 
scribed on  pp.  456,  457. 

This  is  a  manuscript  of  14  folios  on  the  calendar,  and  has  two  dates, 
M.cccc'Mviij"  and  M.cccc°.lxxvij,  on  the  first  folio.  It  also  has  some 
lunar  tables  for  1364-1381,  the  treatise  having  probably  been  copied 
about  1 45 8-1 47 7  from  another  one  of  a  century  earlier. 

NICOLO  DE  ORBELLI.  Latin  MS.,  1478. 

Title.  'Incipit  op9  fratris  (.?)  dorbelli  fup  .  .  .'  (F.  i,  r.).  '  In- 
cipit.  Matematica  '  (F.  132,  r.). 

Colophon.  On  f .  1 26,  v. :  '  Explicimt  Scripta  f fs  nicolai  de  orbel- 
lis  //  doctoris  eximini  fup  .  .  .  //  Deo  gratias  et  xp6  ihu  Amen 
//M?478.' 

Description.  6°,  10.5  x  15.6  cm.,  the  text  being  9.2  X  1 1.9  cm. 
270  ff.  (6  blank),  41  11.     Latin  MS.  on  ]-)aper,  1478. 


474  RARA   ARITHMETICA 

The  first  part  of  the  book  is  devoted  to  dialectics  and  logic.  The 
mathematics  begins  on  f.  132,  r.,  and  consists  of  two  folios  on  the  theory 
of  numbers  and  three  on  geometry.  On  f.  140,  r.,  begins  a  treatise  on 
philosophy,  containing  (f.  184,  v.)  a  drawing  of  '  Johs  fcotus  docts 
fubtihs.'  On  f.  193,  r.,  begins  a  treatise  on  astronomy:  '  Hie  //  Incipit 
liber  de // celo  &  mundo,'  The  rest  of  the  work  relates  to  science  in 
general.    (See  p.  23,  1485.) 

LEONARDUS  MAYNARDUS.  Latin  MS.,  c.  1488. 

A  fifteenth-century  mathematician  of  Cremona,  Italy. 

Title.  '  Leonardi  Cremonenfis  artis  metri//ce  pratice  compi- 
latio.     Primus  tractatus.'     (F.  i,  r.) 

Description.  4°,  17.2  x  24.4  cm.,  the  text  being  7  x  10.8  cm. 
24  ff.,  25  11.,  figures  on  the  margins. 

This  is  a  treatise  on  trigonometry,  and  has  been  included  in  this 
catalogue  because  of  the  arithmetical  nature  of  some  of  the  problems. 

Favaro's  careful  investigation  of  the  time  when  Maynard  lived  still 
leaves  the  matter  in  doubt.  He  may  have  lived  in  the  latter  part  of  the 
fifteenth  century,  or  he  may  possibly  have  been  the  'Leonardus  de 
Antoniis  de  Cremona,  ordinis  minorum,  bacalarius '  who  lived  early  in 
the  fifteenth  century. 

In  the  catalogue  of  the  Boncompagni  sale  another  Latin  manuscript 
of  Leonardo  is  given  as  'in  pelle  di  33  carte  membranacie  del  secolo 
XIV.'  If  this  is  correct,  which  is  doubtful,  Leonardo  must  have  lived 
before  1400.  See  also  Enestrom  in  Bibliotheca  Mathematica  IV  (3), 
p.  290,  and  Favaro  in  the  same  journal  IV  (3),  p.  334.  The  latter  gives 
a  bibliography,  with  some  quotations. 

In  1902  M.  Curtze  translated  and  edited  one  of  the  three  manu- 
scripts of  this  work  known  to  him.  This  is  in  the  Venetian  dialect, 
and  belongs  to  the  University  library  at  Gottingen.  He  also  consulted 
the  two  Latin  manuscripts  formerly  belonging  to  the  Boncompagni 
library,  of  which  this  is  the  later  by  a  few  years.  This  manuscript 
belonged  to  the  advocate  Cav.  Carlo  Morbio,  in  Milan,  before  Bon- 
compagni secured  it.  A  note  on  f.  15,  r.,  shows  that  in  1655  it  belonged 
to  Bonifacio  or  Josefifo  Aliprandi.  Another  note,  on  f.  i,  refers  to  a  pas- 
sage in  a  work  by  Franciscus  Arisius,  printed  at  Parma  in  1702.  This 
passage  is  as  follows:  '  LXXXVIII  (i.e.  1488)  Leonardus  Maynardus 
Insignis  Astronomus,  Physicus  et  Mathematicus,  cuius  opusculum  M.  S. 
Mediolani  servatur,  mihi  indicatum  ab  eruditissimo  Viro  Lazaro  Augstino 
Cotta  I.  C.  amico  meo  nequaquam  satis  laudato,  cui  est  initium  :  (Here 


MANUSCRIPTS  475 

follow  the  first  few  lines  of  tliis  work,  so  nearly  identical  with  the 
manuscript  as  to  show  that  Arisius  probably  had  this  very  one  at  hand.) 
. .  .  Fuit  ante  Blasium  Lconanius  Mayiiardiis,  qui  sue  tempore  non  tan- 
tum  inter  nos,  sed  etiam  inter  omnes  in  lis  studiis  tenuit  principatum.' 
A  similar  passage  ajjpears  in  the  manuscript  Bio^:;rafia  Crcmotiese  of 
Vincenzo  Lancetti,  now  in  the  civic  library  at  Cremona,  from  which  we 
may  infer  that  he  lived  before  Battista  Piasio,  of  a  noble  Cremonese 
family,  a  philosopher,  physician,  and  astrologer,  who  flourished  about 
1500.  This  manuscript  of  Maynardus  is  described  (no.  254)  in  Nar- 
ducci's  catalogue  of  the  Boncompagni  manuscripts  (Rome,  1862). 

ANONYMOUS.  Italian  MS.,  c.  1490. 

Title.  None.  A  manuscript  on  elementary  mathematics. 
'Choncio  fiachofa  che  fono  noue  fighure  nellaba//cho.'    (F.  i,  r.) 

Description.  4°,  14.5  X  21.5  cm.,  the  text  being  9  X  15.2  cm. 
158  ff.  (5  blank),  26  11. 

The  arithmetic  is  of  the  ordinary  commercial  type,  and  includes  the 
fundamental  operations  together  with  a  considerable  range  of  business 
applications.  It  follows  the  general  style  of  the  Florentine  arithmetics 
and  uses  the  Tuscan  herichiiocholo  instead  of  the  Venetian  scachero  to 
designate  our  present  multiplication  :  '  Volendo  multiprichare  .2.  numeri 
p  berichuocolo  '  (f.  17).  The  dates  in  the  problems  in  equation  of  pay- 
ments indicate  that  it  was  written  about  1490  (ff.  83-86-129).  The 
per-cent  symbol  %  here  appears  as  p  c,  p  c",  p  <^°,  as  well  as  p  1 00 
(fif.  84,  86,  134,  V.    See  Fig.  237). 

The  second  part  of  the  work  (f.  91)  relates  to  algebra  :  '  Qui  aprffo 
fcriuerio  lareghola  dellarabre  (dell'  algebra),  m^'ochabiln"  (e  muqabala).' 
The  work  is  rhetorical,  there  being  practically  no  symbolism  employed. 
This  part  of  the  work  closes  with  the  words,  '  Voglia  hora  fare  fine 
enondire  piu  fopa  a  que//fta  reghola  delagebrem"ghabile  ...  (f.  109,  r,). 

The  geometry  begins  on  f.  T09,  r.,  and  is  confined  to  simple  men- 
suration. The  work  closes  (ff.  123-153)  with  a  series  of  miscellaneous 
l)roblems. 

ANDREA  DI  GIOVANNI  BATTISTA  LANFREDUCCI. 

Italian  MS.,  c.  1490. 
An  officer  of  the  Republic  of  Pisa  in  1505. 

Title.  None.  A  treatise  on  arithmetic.  'Choncio  fia  chofa 
che  fono  noue  figure  nellabacho.'     (F.  3,  r.) 


476  .     RARA   ARITHMETICA 

cdvuslU  clvn^Lilcnc  frtiifejpituic/^^w  i[4» 

ciJucllo  irmis?  Jtmcictc  \»llo  ftpc^i  AllajpxmA. 
yJvchtzx  caiiclU  Jrfe  ckfi  utcnc  <vxij<^fr  U 

'nclU  ^^  vnUti^fvJj  <^  3 ifcffcn ore  nc(li>  S^ 

QiAJ    Utia  WcTc    C^J)   A-t<3J?  CC(JCA'-L^^<44-tttl1C 

ncl  4tf  .^ct*-*  ^-  »Ticlf.  if  at. Ota  del  mcrimvc  -  ^ot  <iir- 
l^^<f*  «-«  f  wcfi  -^di  4i<>  pC  coc  A'-L<^4t~ 

iicdcr^  ^u^srihp  fxmpj  ccialU^vrna  yatUtadU 

Fig.  237.    From  the  anonymous  manuscript  of  c.   1490 


MANUSCRII'TS  477 

Colophon.  '  Qvefto  Libro.  aitritto.  tlilua.  popia  ajano.  andica. 
Lanfreduccj.'     (F.  96,  v.) 

Description.  4°,  16.8  X  23.4  cm.,  the  text  being  12.1  X  16.5 
cm.    96  ff.  (15  blank),  29  11. 

This  book  is  an  Italian  commercial  arithmetic  of  the  Florentine  type 
written  about  1490.  After  a  brief  treatment  of  notation  the  author 
takes  up  the  '  Librettine  minore,'  or  the  smaller  multiplication  table  to 
10  X  10,  following  this  by  several  examples.  He  then  gives  (f.  9,  r.) 
'  le  libretine  maggiore  '  or  larger  table,  with  multiplication  '  pquadrato  ' 
and  '  p  berichuochulo,'  these  names  showing  the  work  to  be  Tuscan. 
Division  is  followed  by  a  large  number  of  problems,  per  cent  (il  c-^  and 
p  100)  playing  a  large  part.  The  problems  in  ecjuation  of  payments  show 
the  book  to  have  been  written  between  1489  and  1491  (ff.  72-74). 

JOHANN   NEWDORFFER.  German  MS.,  1492. 

A  Niirnberg  Rechenmeister,  c.  14  50- 1500. 

Title.  'HannsDimpfel// (Multiplication  table) //Johann  Ncw- 
dorffer  Rechen-//maifter  vnd  Modift  zu  N.//  1492.'     (F.  i,  r.) 

Description.  8°,  10.5  X  13.9  cm.  11  ff.,  20  11.  Written  on 
vellum. 

This  is  a  beautifully  written  primer  apparently  done  under  the  guid- 
ance of  one  of  the  celebrated  Rechenmeisters  of  the  Newdorffer  family 
of  Niirnberg.  It  was  written  in  1492  and  is  evidently  the  work  of  a 
beginner  in  commercial  arithmetic.  It  opens  wdth  the  addition  and 
multiplication  of  compound  numbers,  and  this  is  followed  by  the  rule 
of  three,  with  some  applied  problems. 

The  name  Hanns  Dimpfel  is  doubtless  that  of  the  pupil  whose  work 
was  done  under  the  direction  of  Newdorffer.  As  to  the  latter,  it  is  not 
improbable  that  he  was  the  father  of  the  celebrated  Schreib-  and  Rechen- 
meister Johann  Newdorffer,  the  founder  of  the  German  calligraphy, 
who  was  born  at  Niirnberg  in  1497,  and  died  there  November  12,  1563, 
and  whose  son  Johann  (b.  February  22,  1543,  d.  October  28,  1581) 
was  also  a  well-known  Rechenmeister. 

ANONYMOUS.  Latin  MS.,  c.  1500. 

Title.    '  De  comutata  pporfie.'    (F.  14.) 

Description.  Fob,  14.5  X  21.5  cm.,  the  text  being-  1 1.5  X  19cm. 
This  is  the  second  part  of  the  1442  manuscript,  ff.  14-32  (p.  458). 


478 


RARA   ARITHMETICA 


This  is  a  Latin  treatise  on  arithmetic,  apparently  written  about  1450- 
1500.  It  relates  almost  exclusively  to  Boethian  ratios  and  figurate 
numbers.  It  closes  with  a  few  pages  on  the  circle,  including  the  follow- 
ing :  '  Rem  noiia  mirabile5.  quadra//tura  ;  circuli  velut  ifcritabilej.// 
apud  doctof  ppli'  olim.  f.  fabile. // pure  cernut  ocuU  vere  demra//bilem 
nuc  i  fine  feculi '  (f.  30,  v.)  ;  '  C.de  quadratura  circuli '  (f.  32,  r.).  The 
table  in  Fig.  238  shows  some  work  in  series  such  as  is  common  in  arith- 


<a. 

.X- 
a. 

.9- 
.  1*-  ■ 

.3-0- 

■  <rr 
7^- 

.  A 

^fcl 

.9- 

■..5- 

'v\ 

? 

■a^ 

■0- 

Ah- 

.19 

.3-(- 

•M- 

.Z7\ 

r 

??. 

Fig.  238.    From  Dc  comutata  pporne,  c.  1500 

me  tics  of  this  period.  Fig.  239  shows  the  multiplication  table  as  com- 
monly seen  in  the  Boethian  arithmetics  (compare  p.  26).  It  also  shows 
(on  line  i)  the  use  of  the  Roman  numerals  in  connection  with  the  Arabic, 
and  (on  lines  9  and  10)  the  absurdly  long  Latin  names  for  ratios. 

ANONYMOUS.  c.  1500. 

Description.  FoL,  14.5  X  21.5  cm.  It  is  bound  with  the  pre- 
ceding manuscript,  but  is  in  a  later  hand.  It  consists  of  ff. 
33-38  of  the  volume. 

The  manuscript  consists  of  an  interesting  set  of  drawings,  including 
one  of  an  astrolabe  (f.  zi)  and  several  horoscopes  (f.  35).  The  latter 
serve  to  fix  the  date  of  this  portion  (c.  1500)  and  the  country  (Hungary) 
in  which  it  was  written.  They  include  (f.  35,  r.)  a  horoscope  cast  at 
the  birth  (1456)  of  Ladislas  or  Uladislas  VII,  possibly  by  Peurbach  (see 
p.  53),  and  one  at  his  coronation  as  king  of  Hungary  on  September  18, 
1490,  mentioning  his  coronation  as  king  of  Bohemia  on  August  25,  147 1, 
and  his  election  as  king  of  Hungary  on  July  15,  1490. 

ANONYMOUS.  Latin  MS.,  c.  1500. 

Descnption.  FoL,  14.5  X  21.5  cm.  It  is  bound  with  the  pre- 
ceding manuscript,  is  possibly  in  the  same  hand  as  the  horo- 
scopes, and  is  certainly  on  the  same  paper.  It  occupies  ff. 
39-44  of  the  volume. 

This  is  a  treatise  on  the  mensuration  of  the  circle  and  dates  from 
c.  1500.  It  has  some  bearing  upon  the  metrical  computations  of  the 
time. 


MANUSCRIPTS 


479 


rtmn/ 

tiUOJj 


c^ntth     KTomn  dcUxtt-  qz  ^f  dL  ot. 


.^■y^; 


rU' 


y  niKi^ 


V   1 1 1 \ 1 ! 1 T f 1 htnihxJc 


4- 


"V 


'S 


^ 


H 


?i  '^ 


2A  \;j- 


}o  Uc 


¥1 


4^ 


4* 


H 


.+> 


cr^ 


-V; 


cr-4 


/\- 


'^^l' 


^6 


/TO 


"ff  )      <j>#/  rt  /iirfu-in'i 
ff"'  L\tT\lr-  .ifint 

|]K|/|(WC    lj  -I  liw/J 

wmhMv-  Tap"  <:  -T^-nniuj  «u)  tietho-  ^.»nr  •  ^  Tdto  yma  c>lJnu    n"V  /-^i  r/t ^x>wl 

Fig.  239.    From  £>e  conmtata  pponie,  c.  1500 


48o  KARA   ARITHMETICA 

The  fifth  portion  of  the  volume  (ff.  45-52)  consists  of  a  set  of  Latin, 
and  the  sixth  (ff.  53-58)  of  a  set  of  Italian  verses.  The  seventh  (ff.  59- 
60)  contains  a  brief  reference  to  astrology.  The  rest  of  the  book  is  of 
a  literary  or  astrological  character.  There  are,  in  all,  thirteen  different 
manuscripts  in  the  volume.  For  details  concerning  the  non-mathematical 
portions,  see  Narducci,  1.  c,  p,  31. 

ANONYMOUS.  Italian  MS.,  c.  i  500. 

Title.    '  Regole  per  far  Orologi  da  sole //con  le  sue  Figure.' 
Description.  4°,  16.8  X  22.3  cm.   22  ff.    Bound  with  the  Verge- 
rius  manuscript  described  on  p.  456,  but  written  in  a  later  hand. 
This  is  an  Italian  treatise  on  dialing,  written  in  a  fine  hand  of  the 
sixteenth  century,   with  well-executed   figures.     Several  other  manu- 
scripts, not  of  a  mathematical  character,  are  also  bound  with  it. 

ANONYMOUS.  Latin  MS.,  1501. 

Title.    None.    A  treatise  on  mensuration. 

Colophon.  There  is  none,  but  one  of  the  folios  bears  the  date 
'  I  501  adj  feptembr  In  Niirnberg.' 

Description.  4°,  15  X  21.8  cm.,  the  text  being  9.8  X  17.1  cm. 
89  ff.  unnumb.  -f  7  blank  =  96  ff.,  27-30  11.  Latin  MS.,  written 
in  a  German  hand.  Bound  with  the  manuscript  of  Mohammed 
ibn  Musa  (p.  454). 

This  is  a  treatise  on  mensuration,  of  no  particular  merit  save  as  it 
shows  the  nature  of  the  work  at  the  opening  of  the  sixteenth  century. 
It  contains  a  copy  of  the  c.  1475  manuscript  described  on  p.  468. 

STEPHANO    DI    BAPTISTA    BELLI    STEPHANI     DA 
MERCATELLO.  Italian  MS.,  Mercatello,  c.  1522. 

An  Italian  teacher,  born  at  Mercatello,  and  living  there  in  1522.     He  was 
a  pupil  of  Paciuolo. 

Title.  'Svmme//Arismetice.'  (F.  i,  r.)  '  Stephano.D.B.Del- 
listpha//ni .  damercatello .  atvtti .  qve//ili .  liqvali .  in  arte .  mercan// 
tile.exercitare.sidilectano.'   (F.  i,  v.) 

Description.  8°,  13.5x20.7  cm.,  the  written  part  being 
9.1  X  14.3  cm.  153  ff.  numbered  (5  blank),  29  11.  Italian  MS. 
on  paper,  except  f.  i,  which  is  on  vellum.  Written,  as  the 
problems  show,  at   Mercatello,  c.  1522. 


MANUSCRII^TS  481 

This  is  a  manuscript  on  commercial  arithmetic,  unusually  < omjilctc 
in  its  appHcations,  and  also  unusually  well  written.  It  is  of  the  general 
Florentine  type,  but,  as  appears  from  a  date  on  f.  loi,  r.,  was  written 
at  Mercatello,  a  town  south  of  Ferrara  and  east  of  Florence.  The  name 
of  the  author  appears  not  only  in  the  dedicatory  epistle,  but  also  at  the 
end  of  a  bill  of  exchange  dated  '1522  A  di./ 25/maggio  in  M"° ' 
(Mercatello),  in  the  latter  case  as  '  Stefano  di  Bap'^  Stefanj  fj.'  The 
examples  in  equation  of  payments  are  dated  1371-72,  showing  that 
these  were  copied  from  some  earlier  work,  and  in  fact  they  were  taken 
from  Paciuolo  ('  Diftinctio  nona,  tractatus  cjuintus '),  who  in  turn 
borrowed  them  from  some  predecessor. 

Stephano  states  in  his  dedicatory  epistle  that  he  was  a  pupil  of 
Paciuolo,  and  that  he  is  chiefly  indebted  to  him  for  his  material :  '  et 
maxime  dal  mio  R''".  et  ex'^'  affme  &  pceptor'.  M.  Luca  dal  Borgo.' 
This  is  quite  evident  on  comparing  certain  passages  ;  for,  while  Stephano 
does  not  usually  copy  his  master  verbatim,  there  is  often  a  great  simi- 
larity between  them,  and  sometimes  (as  in  the  chapter  '  De  le.  2.  falfe 
pofitioni ')  there  is  evident  plagiarism.  Stephano,  however,  omits  most 
of  the  theory  of  numbers  to  be  found  in  Paciuolo,  and  confines  himself  to 
mercantile  applications.  These  cover  barter,  partnership,  various  forms 
of  discount  and  exchange,  and  other  similar  topics.  There  are  also  given 
a  number  of  mediaeval  puzzles,  including  the  testament  problem,  the  sale 
of  the  eggs,  the  hound  and  hare,  and  the  guessing  of  numbers.  A  little 
work  in  mensuration  and  the  calendar  is  given  at  the  end  of  the  book. 

BARTOLOMEO  ZAMBERTO,  editor.       Latin  MS.,  c.  1525. 

A  Venetian  scholar  of  c.  1500.     He  was  born  c.  1473. 

Title.  '  Evclidis  //  Megarenfis  graeci  philofo-//phi  ex  Theone 
graeco  com-//metare  Interprete  Zaber-//to  veneto  triplex  prici- 
pioru  //gentis  pn'mu  diffinitiones:  //  Signv//est  cuius  pars  nulla: 
//  Linea  vero,  lon-//gitudo  ilia  tabilis.//  Lineae  autem  limites,  funt 
figna. // Recta  linea,  est  quae  ex  aequa-//li  fua  interiacet  figna. 
Superfi-//cies,  est  quae  longitudinem  latitu//dinemq5  tantuni 
habet.  Super-//ficiei  extrema,  funt  lineae.  Pla-//na  fuperficies, 
est  quae  ex  aecjua-Z/li,  fi-ias  interiacet  lineas.'     (F.  2,  r.) 

Description.  12°,  8x  11.7  cm.  152  ff.  (2  blank),  16-20  11. 
Written  on  parchment. 

This  beautifully  written  manuscript  has  been  included  in  this  cat- 
alogue  of   arithmetics   without   much   justification,   since   it   does  not 


482  RARA   ARITHMETICA 

contain  even  Book  V.  It  is,  however,  valuable  in  showing  the  influence 
of  printing  upon  written  numerals.  It  is  a  copy  of  a  translation  of  the 
first  three  books  of  Euclid  made  by  Bartolomeo  Zamberti  of  Venice  in 
1513.  The  complete  translation  was  first  printed  in  Venice  in  15 13, 
the  statement  of  some  bibliographers  that  it  appeared  in  Paris  in  1505 
being  unsupported  by  any  evidence.  It  also  appeared  in  Venice  in 
1517,  in  Paris  in  1516,  and  in  Basel  in  1537,  1546,  and  1558.  See 
Riccardi,  II,  i,  644;  Weissenborn,  Die  Uebersetzungen  des  Euklid 
durch  Campano  taid  Zamberti,  Halle,  1882.  This  manuscript  was  evi- 
dently written  about  1525  for  some  noble  family,  for  it  has  (f.  2,  r.) 
three  illuminated  coats  of  arms, 

ANONYMOUS.  Latin  MS.,  1533. 

Title.  '  Declaratio  Calendarii  et // Almanach  huius  Cifte.' 
(F.  I,  r.)    '  Ars  supptitandi  cum  Denariis.'    (F.  66,  r.) 

Description.  4°,  16.6  X  23.3  cm.,  the  text  being  11.8x16.3  cm. 
81  ff.  unnumb.,  18  11.     Latin  MS.,  written  on  vellum,  in  1533. 

This  beautifully  written  Latin  manuscript  consists  of  two  distinct 
works.  Of  these  the  first  is  a  computus  in  twenty-three  chapters,  written 
apparently  in  Salisbury  cathedral  in  1533.  The  second  part  is  a  treatise 
on  counter  reckoning,  and  consisted  originally  of  six  chapters,  '  De 
Numeratione,  Additione,  Subftractione,  Multiplicatione,  Diuifione,  Frac- 
tione  minutiis.'  The  last  of  these  chapters  is  missing.  The  manuscript 
is  particularly  interesting  because  it  gives  the  counter  reckoning  as  it 
was  used  in  England  early  in  the  sixteenth  century,  the  numbers  all 
being  written  in  Roman  when  they  are  not  represented  '  on  the  line.'  It 
is  illustrated  by  numerous  diagrams  representing  the  line  abacus.  The 
manuscript  closes  with  five  pages  '  De  proportione  vel  regula  Detri,' 
and  '  De  Proba  regule  Detri.' 

ANONYMOUS.  Italian  MS.,  c.  1535. 

Title.    'Trattato  d'Aritmetica,  e  del  Misure.'     (F.  i,  r.) 
Description.   FoL,  18  X  23.8  cm.,  the  text  being  14.5  X  21  cm. 
148  ff.,  23-27  11. 

This  is  an  Italian  manuscript,  written,  as  the  dates  on  folios  67,  68 
show,  about  1535.  It  is  a  commercial  treatise,  beginning,  as  was  fre- 
quently the  case,  with  the  fundamental  operations  with  compound 
numbers.  The  method  of  division  '  a  danda  '  is  preferred  to  that  '  per 
galea,'  although  both  are  given  :   '  II  partir  a  galea  e  molto  legiadro  et 


MANUSCRIPTS 


483 


fpeditiuo,  ma  non  tanlo  ficuro  per  un  ])riii(i|)iantL'  ([uanto  il  partir  a 
danda'  (f.  19,  v.).  'I"he  apjjlied  problems  are  generally  of  a  practical 
type  useful  to  merchants'  apprentices  in  the  north  of  Italy.   (Fig.  240.) 


Qipm^9tC^.  Sut.  34.  (); 
(|]«pi»Bf?9ffx  Air  X  }  o. 
(af»if8^  9ff  ;*  Pur     -i.  4  -u- 


S    I   -I.  0 


8b- 


^umn">f -^^ovfi  .  5 


J     3     T.  c; 


%  4  ^ 


Fig.  240.    P'rom  the  anonymous   Trattato  ok  c.  1535 

Much  of  the  latter  part  of  the  treatise  relates  to  mensuration.  A  set  of 
tables  and  some  notes  are  given  at  the  end.  Among  the  notes  is  the 
celebrated  problem  of  the  horseshoe.  (Fig.  241.) 


484  RARA   ARITHMETICA 

A  ^^^  ok   ^vi^^   c/:<.^'^^  ^  ^u^c^^/   j^^  V<^  ^.^ 

.  y  ?  'tr/c      A/^x«/«s  '^'^^y  c,  jc^A  A^  /^^  /»u  2^  e 


n.0  'ir'tir- 


"^^ ^^ —  S  >o  ^:  ^;^;  r^ 

Fig.  241.    Notes  in  the  anonymous  manuscript  of  c.  1535 

The  manuscript  is  no.  23  in  Narducci's  catalogue  of  the  Boncom- 
pagni  manuscripts  (Rome,  1862,  p.  16). 

LUDOVICO  ALT  DI   SALISPURGO.       Italian  MS.,  1545- 

A  sixteenth-century  student. 

Title.   '.M  .D.  XLV.//  Ludouicho  alt  de  Salifpurga.'   (F.  i,  r.) 
Description.    8°,  12.6  X  17.8  cm.     92  ff. 


MANUSCRIPTS 


485 


This  is  a  business  arithmetic  written  by  some  stmlent  in  1545.  'I'he 
author  first  treats  of  the  fundamental  operations  with  denominate  num- 
bers, following  this  by  a  treatment  of  fractions.    (Fig.  242.)   The  last  part 

^5%l..lii,V.  ^^^l  ^*  ^'^^  ru^tc^  .  (toe  oPCo  x^ 


'ciu 


Fic.  242.    From  the  ludovico  alt 

of  the  work  is  a  '  praticha '  (f.  48),  and  considers  the  ordinary  business 
arithmetic  of  the  time.  The  author  uses  almost  the  same  form  of  the  per- 
cent symbol  as  the  writer  of  1456  referred  to  on  p.  458,  viz.  p  ^  (see 
Fig.  243).    The  manus(  ript  is  described  in  Narducci's  catalogue,  p.  152. 


486 


RARA   ARITHMETICA 


^<^ 


#  l\ 


^   ^^- 


1  /^  fl  * 


A 


^il^^l,.^'^?^-?'^^-^' 


Fig,  243.    From  the  ludovico  alt 

ANONYMOUS.  Latin  MS.,  c.  1550. 

Title.    '  Introducoriu  breue  fup  elementa  Euclidis.' 
Description.    4°,  15  X  21.8  cm.,  the  text  being  14.2  X  19  cm. 
23  ff.  unnumb.  +  2  blank  =  25  ff.,  28-33  H-    Written  in  a  Ger- 
man hand,  c.  1550. 


MANUSCRIPTS 


487 


This  manuscript  is  the  last  one  in  the  volume  containing  the  algebra 
of  Mohammed  ibn  Musa  (p.  454).  It  consists  o(  introductions,  book  by 
book,  to  books  I-XV  of  Euclid.  The  symbols  +,  ^  (for  -),  ^T",  and 
MSf     (for  ^  )  are  used. 

IIONORATUS.  Latin  MS.,  c.  i  550-1600. 

A  Venetian  monk  of  the  sixteenth  century. 
Title.    'Optts   Arithnicticfi    D.    Ilonorati    vcnc-//t-i    nionachj 
cocnobij  S.  Latirclij.'     (F.  i.) 


Fk;.  2. 


Yuo-si   HONOR.VrUS 


Description.  8°,  10.9  x  16  cm.,  the  text  being  about  9.5  x 
15.8  cm.     Ill  ff. 

This  manuscript  was  written  by  a  pupil  of  a  \'enetian  monk  named 
Honoratus  evidently  about    1 550-1 600.     It  is  a  practical  arithmetic, 


488  KARA   ARITHMETICA 

the  author  taking  up  rather  fully  the  fundamental  operations  with  inte- 
gers and  denominate  numbers.  He  also  treats  of  such  common  applica- 
tions as  partnership  and  barter.  The  illustrations  are  those  which  a 
pupil  would  be  likely  to  make,  and  prove  the  manuscript  to  be  the  work 
of  an  immature  hand.  The  common  galley  method  of  division  is  given, 
with  the  ship  in  full  form  (Fig.  244),  and  the  sign  %  appears  as  per 
cr-°  (see  p.  439). 

ANONYMOUS.  Italian  MS.,  c.  1560. 

Title.  '  Dichiarazione  per  intelligentia  de  Principiantj  //  del 
vso,  che  merchantilmente  tiene  la  Citta  dj  Firenze  //  sopra  le 
monete,  pesi,  e  Misure.'  (F.  2,  r.)  On  f.  i,  r.,  the  coat  of  arms 
of  the  Angelotti  (.?)  family  is  painted  on  parchment. 

Description.  Fol.,  18.7  X  25.5  cm.,  the  text  being  12.5  x  20 
cm.     166  ff.  (3  blank),  8-42  11. 

This  is  a  very  clearly  written  Florentine  manuscript  on  commercial 
arithmetic.  The  8's  are  all  made  like  S  on  its  side,  thus  (c/D),  a  feature 
not  infrequently  seen  in  the  second  half  of  the  sixteenth  and  even  in  the 
seventeenth  century.  As  was  often  the  case,  the  author  presupposed  a 
knowledge  of  the  fundamental  operations,  reviewing  them  only  briefly 
with  compound  numbers.  The  problems  are  of  the  common  mercantile 
type  of  the  period.  The  manuscript  seems  to  have  been  written  about 
1560.    (See  Fig.  245.) 

ANONYMOUS.  Latin  MS.,  c.  1565. 

Title.  'Trattato  d'aritmetica  mercantile.'  (Cover.) 
Description.  Sm.  4°,  1 1.9  X  16.8  cm.,  279  ff.,  20-25  11. 
It  appears  by  the  problems  that  this  mercantile  arithmetic  was 
written  at  Bologna  about  1565.  Like  so  many  similar  treatises,  it  opens 
with  a  set  of  column  multiplication  tables.  The  first  operation  is  '  par- 
tiri  piccoli,'  or  short  division,  this  being  necessary  for  the  simple  reduc- 
tions in  the  addition  of  denominate  numbers.  This  is  followed  by  the 
addition  of  pounds,  shiUings,  and  pence  ('Sommare  di  lire  e  fs  e  de'). 
The  next  operation  is  multiplication,  several  methods  being  given.  The 
first  is  cross  multiplication  ('  modo  di  mcare  p  +  ')  !  the  second,  from 
left  to  right  ('  modo  di  multrip''S  per  la  dirieto,'  multiply  always  appear- 
ing as  '  multriply,'  '  multripich,'  in  this  manuscript)  ;  the  third,  by  ref- 
erence to  the  tables  ('  p  colonna ')  ;  the  fourth,  our  common  form,  the 
'  bericuocolo  '  of  the  Florentines,  but  here  called  '  modo  per  bilicuocolo  ' 


MANUSCRIPTS  489 

\  no  ^la  n'cearo  d'^nv^rzssf  fi  opbi_ib-cr, 
pzvo  a  ragione  di^y  ^J'^'*  ^-oo-sj  domaru 
cLa  ciun^ue  quanro  joi  il  JTJndo. 


xoo 00 

1000 


7-^ -100 o>fr.jfco' 

I^       ^'. 

C/y  ..-•4- 


> 


070  000      —        — 

(To  o  o    —     - 

100      — 

Cyo     - 

3  —  6^—00 


CofzcOS-  f-   CO 4: 

3  1  /XIIQ.0:^  > 


3  7  •■ 


IJ 2^0 


^^B 


Fin.  245.    From  an  Italian  manuscript  of  c.  1560 


ir 


490  RARA   ARITHMETICA 

(also  '  bilicocolo ').  This  is  followed  by  the  subtraction  of  denominate 
numbers,  and  this  by  a  more  extended  treatment  of  multiplication. 
Division  follows,  at  first '  per  ripiegho,'  and  then  '  per  danda'  (Fig.  246), 
the  galley  method  not  being  given.  This  is  followed  by  a  treatment  of 
fractions,  percentage  (' Commicano  e  centi  dalcuna  merchantia '),  the 
rule  of  three  ('  Commica  lareghola  del  tre  '),  profit  and  loss,  equation  of 
payments,  exchange,  and  the  other  mercantile  rules  of  the  period.  Alto- 
gether it  is  one  of  the  best  of  the  sixteenth-century  manuscripts  on  the 
commercial  arithmetic  of  Northern  Italy. 

GAUDIOSO  FRASCADA.  Italian  MS.,  1568. 

A  schoolmaster  of  Brescia,  about  the  middle  of  the  sixteenth  century. 

Title.    '  Libro  di  Arithmetica //et  Geometria.'     (F.  i,  r.) 

Colophon.   '  II  j5nte  libro  cioue  opera  di  abaco  et  geometria  scrit- 

to  ad  //  instantia  de figliolo  di  m  Bertholameo  sachetto  // 

habitante  nella  terra  de'  I'orala  algise'  scritto  per  mi  //  Camillo 
sachetto  I'anno  1568  a  honor  de  dio  et  della // iiergine  maria  laus 
deo  '  (f.  43,  v.).  But  f.  5,  v.,  has  the  following:  ;  Questo  libbro 
e  ftato  fcritto  p  Gaudioso  frafcada  cittadino  di  //  Bressa  a  di 
4  Nouembrio  lanno  1555  laus  dei.' 

Description.  FoL,  20.2  x  28.6  cm.,  the  text  being  14.7  x  24.2 
cm.    46  ff.  (3  blank),  17-29  11. 

This  is  a  copy,  made  in  1568,  of  a  treatise  composed  by  Frascada, 
a  schoolmaster  of  Brescia,  in  1555.  The  author  uses  both  the  Floren- 
tine and  the  Venetian  forms  and  names  for  the  operations,  as  in  the 
expression  '  Multiplicar  per  fchacchiere  feu  baricocolo.'  Several  forms 
of  multiplication  are  given,  as  in  the  work  of  Paciuolo,  but  in  division 
only  the  galley  method  appears.  The  examples  are  generally  of  the 
ordinary  business  nature,  and  the  rules  include  *  Raggioni  p  lo  cattayno, 
cioe  position  false.'    The  last  few  folios  refer  to  practical  geometry. 

ANONYMOUS.  German  MS.,  c.  1575. 

Title.  '  Von  KiiftHcher  Abmeffung  aller  groffe,//ebene,  oder 
nidere,  in  die  lenge,  hohe,  breite  vnd//tiefte  als  graben  Cifternen, 
vnd  Brunnen.'    (F.  i,  r.) 

Description.  12°,  10.2  x  15  cm.,  the  text  being  8.6  x  13  cm. 
30  ff.  unnumb.  -f  4  blank  =  34  ff.,  24  11.  Written  in  a  German 
hand  and  in  the  German  language,  c.  1575. 


MANUSCRIPTS 


491 


<^ax^C'yn"^^'>^^7Sig>o\t^M 


'2±::isy 


^oo.  --^|. 


0-0.3.  •  -'"  •  t 

It  6$  8 


if^ 


Fig.  246.    From  an  Italian  manuscript  ok  c.  1565 


492  RARA   ARITHMETICA 

This  manuscript  contains  a  few  numerical  problems  in  connection 
with  mensuration.  The  work  is  of  no  merit  save  as  it  throws  light  upon 
the  mensuration  of  the  sixteenth  century  in  Germany.  Like  most  books 
of  this  kind,  written  at  that  time,  it  has  little  or  no  explanation  of  the 
rules  used. 

DOMENICHO  DA  BIEN  DE  VALSUGANA. 

Italian  MS.,  1579. 

A  Venetian  student  of  1579. 

Title.    None.    An  elementary  algorism. 

Colophon.  On  p.  21  there  is  this  statement:  '  L'anno.  1579- 
a  di  .9.  del  //  mese  di  marzo  in  liedolo  //  Per  Domenicho  da  Bien 
de  Valsugana.' 

Description.  12°,  9.5  X  14.5  cm.,  the  text  being  about  7.5  x 
12.5  cm.  48  pp.,  varying  number  of  lines.  Probably  written  at 
or  near  Venice.     1579. 

This  is  a  student's  manuscript  on  commercial  arithmetic.  It  opens 
with  the  fundamental  operations,  the  treatment  of  '  Multipticare  Per 
scachiero  '  (p.  12)  showing  the  Venetian  instead  of  the  Florentine  influ- 
ence. Division  is  performed  only  '  per  galia,  ouero  per  battello  '  ('  p 
Galera'  Domenicho  calls  it  elsewhere),  or  'per  Collona,'  the  modern 
form  not  appearing.  The  examples  are  wholly  mercantile,  most  of  the 
applications  being  in  the  rule  of  three.  The  work  also  contains  some 
simple  examples  in  mensuration. 

FRANCESCO  GIUNTINI.  Italian  MS.,  Lyons,  1579. 

An  Italian  student  or  teacher  of  astronomy.    I  judge  from  his  horoscope 
that  he  was  born  November  14,  1522,  at  Florence. 

Title.    None. 

Colophon.  'Di  Lione  adi  13.  di  Maggio  1579:  Francesco 
Giuntinj.' 

Description.  4°,  15.5  X  21.8  cm.,  32  ff.  (2  blank),  26-30  11. 
Written  on  paper. 

This  manuscript  is  clearly  written  in  Italian,  and  relates  to  astrology, 
in  particular  to  Giuntini's  own  horoscope.  Its  interest  in  the  history  of 
arithmetic  lies  wholly  in  such  symbols  as  that  for  degrees,  and  in  the 
forms  of  numerals. 


MANUSCRIPTS  493 

JOHANNES   KLUMPIUS.  Latin  MS.,  1598-99. 

Title.  '  Varij  Tractatus  Mathematices //a  Joanne  Klum])i(> 
philfse/Zftudiofo  excpt  Ingol-//ftadij  Anno // i  598  ct  fequcnlj.' 
(P.  I.) 

Colophon.  At  the  end  of  the  '  Arithmetica  i)ractica  '  are  the 
words  'Abfokum')  15  Januarij // Anno  1599.' 

Description.  4°,  15.3  X  20  cm.,  the  text  being  10.5  x  16.5  cm. 
(varying).  70  pp.  blank  +  442  pp.  written  =512  pp.,  19-21  11. 
Latin  MS.,  written  in  a  German  hand,  Ingolstadt,  1598-99. 

This  is  a  set  of  lecture  notes  on  general  mathematics,  including  geom- 
etry, trigonometry,  arithmetic,  and  astronomy.  The  arithmetic  consists 
of  two  distinct  parts,  probably  the  result  of  two  courses  of  lectures. 
The  first  treats  of  the  four  fundamental  operations  with  integers  and 
common  fractions  ;  the  second  of  practical  arithmetic,  as  Klumpius  calls 
it,  although  it  simply  gives  the  fundamental  operations,  j>rogressions, 
roots,  and  a  few  rules  like  the  rule  of  three,  partnership,  alligation, 
and  the  rule  of  false. 

B.   ROTH.  German  MS.,  1599. 

A  German  student  of  c.  1600. 

Title.  '  Das  Flinfft  Capitel.//  Item  Im  FunlTten  Capitel  ift 
von  dem  Algorifmo //oder  Cofs.'     (F.  4,  r.) 

Colophon.  Not  entirely  legible.     It  contains  the  date,  August  4, 

1599- 

Description.  4°,  18.8  X  24.6  cm.,  the  text  being  13.5  X  19.8 
cm.     321  ff.  (13  blank),  the  lines  varying. 

This  is  a  German  manuscript  containing  the  solutions  of  the  problems 
in  Stifel's  edition  of  Rudolff's  Coss,  beginning  with  Chapter  5.  (See 
p.  258.)  This  manuscript  was  written,  as  the  colophon  shows,  in  1599. 
The  name  of  the  writer  is  not  entirely  legible,  but  it  seems  to  be  Brite- 
nus  Roth.  The  solutions  are  written  in  a  very  clear  hand,  and  furnish 
an  excellent  example  of  the  symbolism  of  that  period. 

ANONYMOUS.  Dutch  MS.,  Louvain,  c.  1600. 

Title.    None.    A  treatise  on  mensuration. 

Description.  Sm.  8°,  10.5  x  16  cm.,  the  text  being  about  8  x 
13  cm.    80  pp.,  33-46  11.    Written  on  vellum. 


494  RARA   ARITHMETICA 

This  is  a  Dutch  manuscript  on  gauging  and  general  mensuration. 
The  gauger's  tables  are  as  clearly  written  as  in  the  best  Florentine 
manuscripts. 

ANONYMOUS.  Latin  MS.,  c.  1600. 

Title.    None.    A  treatise  on  arithmetic. 

Description.  Sm.  4°,  10.5  X  14. i  cm.,  the  text  being  about 
6x12  cm.  197  ff.  numb,  (there  have  been  added  8  ff.  in  the 
back,  written  on  vellum,  making  a  total  of  205  ff.),  22  11.  (varies). 
Latin  MS.,  written  in  a  German  hand. 

This  manuscript  begins  with  the  fundamental  operations  after  the 
manner  of  algorism,  and  then  takes  up  the  theory  of  numbers  according 
to  the  Boethian  system.  Figurate  numbers  and  proportions  (ratios)  are 
treated  at  great  length,  as  in  the  arithmetic  of  Boethius,  although  this 
is  not  a  copy  of  that  work.  It  is  not  common  to  find  manuscripts 
written  as  late  as  this  that  go  so  fully  into  the  ancient  theory  of  pure 
arithmetic.  At  the  end  of  the  book  the  author  has  placed  eight  folios 
of  '  Arithmetica  tabulais  et  formularis,'  clearly  written  on  vellum,  con- 
taining multiplication  (or  *  area  ')  tables  of  primes,  of  '  oddly  even  '  and 
'  evenly  odd '  numbers,  of  '  solid  numbers,'  and  the  like. 

ANONYMOUS.  German  MS.,  c.  1600. 

Title.   'Niitzlicher  Gebrauch//Der  Weldt  Kugel.'     (P.  3.) 
Description.  FoL,  20.5  x  33  cm.,  the  text  being  19.5  x  27  cm. 
(varies),  160  pp.  (several  blank),  24-29  11.  (varies).    Written  on 
paper,  in  German,  c.  1600. 

Although  the  first  part  of  this  manuscript  is  on  the  terrestrial  sphere, 
the  second  part,  beginning  on  p.  83,  is  on  arithmetic  as  needed  by  cos- 
mographers.  This  includes  the  fundamental  operations,  including  square 
and  cube  roots.    The  galley  method  of  division  is  used  exclusively. 


ADDENDA 

Since  the  completion  of  the  manuscript  for  the  edition  tie  luxe  of 
this  work,  two  years  ago,  numerous  additions  have  been  made  to  Mr. 
Plimpton's  collection  of  early  arithmetics.  Excluding  a  number  of 
early  Arabic  manuscripts,  these  acquisitions,  with  references  to  the 
pages  on  which  they  would  naturally  appear,  are  as  follows  : 

Page  i6.  The  1561  edition  of  Borghi  has  been  acquired. 

Page  23.  The  1485  edition  of  Nicolo  de  Orbelli  has  been  acquired. 
See  also  page  473. 

Page  36.  There  has  been  acquired  a  work  on  the  calendar  by 
Regiomontanus  :  'Almanach  magiftri  Johanis//de  monteregio  ad  anos. 
xviij.//acuratiffu-ne  calculata.'  Colophon  .-...'  Erhardi  Ratdolt  Auguf- 
ten  // Vindelico  .  .  .  M.cccc.lxxxviij.'  This  is  not  an  arithmetic,  but  it 
is  interesting  on  account  of  its  mathematical  treatment  of  the  calendar. 

Page  62.  There  has  been  acquired  a  Cracow  edition  of  a  work  by 
Faber  Stapulensis,  containing  a  little  arithmetic :  '  Jacobi  Fa-//bri 
Stapv-//leii  T  Artiu  diuifione  itroductio  .  .  .'  Colophon  :  '  Cracouig  .  .  . 
1534.'    4°,  13.8  X  18.7  cm.,  the  text  being  10.2x16  cm. 

Page  82.  The  first  (1503)  edition  of  the  Margarita  Philosophica  has 
been  acquired.  The  colophon  is  as  follows  :  '  Chalchographatum  primi- 
ciali  hac // peffura/ Friburgi  p  loanne  Scho//ttu  Argeii.  citra  feftu 
Margareth^  //anno  gratis  M:CCCCC.III.' 

Page  114.  An  edition  of  Tagliente's  '  Libro  //de  abaco,'  '  Vinegia  . . . 
M.D.XLIIII'  has  been  acquired.  Also  the  Venice  edition  of  1570,  and 
an  edition  s.  a. 

Page  115.  The  1570  Milan  edition  of  Tagliente  has  been  acquired  : 
'  Libro  //  de  Abbaco  che  inse-//gna  a  fare  ogni  ragione  //  mercadantile 
.  .  .  Milano  1570.' 

Page  139.  The  1525  edition  of  Riese's  first  book  has  been  acquired  : 
*  Rechnung  auff  der  linihen  //gemacht  durch  Adam  Riefen  vonn  Staff el- 
//fteyn/in  maffen  man   es   pflegt   tzu  lern  in   alien //rechenfchulen 


RARA   ARITHMETICA 

gruntlich  begriffen  anno  15  i8.// vleyfigklich  vberlefen/ vnd  zum  andern 
mall //in  trugk  vorfertiget.//C,Getruckt  zu  Erffordt  zcum//Schwartzen 
Horn.// 1525.'  Co/(?/>/iofi :  ^HGedruckt  tzu  Erffordt/ durch//Mathes 
Maler.  M.//CCCCC.XXV.  Jar.'  8°,  8.8  x  14.6  cm.,  the  text  being  7.1  x 
10.7cm.    44ff.,  17-20II. 

Page  139.  Another  Erfurt  edition  of  Riese's  second  work  has  been 
acquired:  '  Rechnung  auff//der  Lynihen  vii  Federn///Auff  allerley 
handthirung/ gemacht  durch//Adam  Ryfen.//  Zum  andern  mal  vber- 
fehen///vnd  gemehrt.// Anno  M.  D.  XXvij.'  Colophon:  '  Gedruckt 
zu  Erffurdt  zum  Schwartzen  Horn.' 

Page  148.  The  1570  Venice  edition  of  Feliciano  has  been  acquired. 

4  +  79ff. 

Page  167.  'Ulrich  Kern  von  Freysing  Eyn  new  Kunftlichs  wolge-// 
grundts  Vifierbuch/  gar  grvifl  vnnd  behend  //  aufz  rechter  art  der 
Geometria/. . .  M.D.XXXI.'  FoL,  18  x  27.8  cm.,  the  text  being  12.7  x 
23  cm.    57  ff.,  46  11.    Strasburg,  1531.    A  work  on  guaging. 

Page  181.  Two  more  editions  of  Mariani's  '  Tar  iff  a '  have  been 
acquired,  Venice  1564  and  1572. 

Page  223.  There  is  mentioned  an  arithmetic  by  Medlerus,  1543, 
with  a  second  edition  in  1550.  This  work,  recently  acquired,  may  be 
all  that  there  is  of  the  1550  edition:  '  Facili-//ma  et  exactis-//sima 
ratio  extra//hendi  radicem  Quadratam//&  Cubicam,  a  Doctore// 
Nicolao  Medlero  in  //  gratiam  ftudiofae  //  iuuentutis  //  sedita,  //  Anno 
Domini. // M.D.L.'  Colophon:  '  Impressvm  VVitem- // bergae,  per 
Vi-//tum  Creutzer.'  8°,  9  x  14.8  cm.,  the.  text  being  7  x  11.2  cm. 
7ff.,  27  11. 

Page  322.  The  1575  edition  of  Lapazzaia  has  been  acquired.  It  is 
not  the  same,  however,  as  the  work  described  on  page  322,  either  in 
title  or  in  contents.  '  Opera //terza//de  aritmeti-//ca  et  geo-//metria. 
//Deir  Abbate  Georgio  Lapazaia//da  Monopoli.  //Intitolata  il  Rama- 
glietto.//In  Napoli // Apreffo  Mattio  Cancer.  M.D.LXXV.'  14.2  x  20 
cm.,  the  text  being  II  X  16  cm.   4 -f  169  pp.,  29  11. 

Page  322.  The  1566  edition  of  Lapazzaia  has  been  acquired  :  *  Fami- 
liarita // d'arithmetica,  e  geometria // con  I'vsitata  prattica  Napo-// 
litana,  Compofta  &  ordinata  per  Abbate  Geor-//gio  Lapizzaya  Canon ico 
Monopolitano // Nuouamente  con  fomma  diligentia//Riftampata,  e 
corretta.//  ...  In  NapoH  //Appreffo  Horatio  Saluiani  //MDLXVI.'  4°, 
14  X  22  cm.,  the  text  being  11. 2  x  16.2  cm.    2  +  62  ff.,  28  11. 


ADDENDA 

Page  340.  '  L'Arithmeti(ive//milit;iire  d'AlexantireZ/Vandenbussche 
Flandrois//departie  en  deux  liures.//.  .  .  A  Paris  .  .  .'  (s.  a.).  8°,  14.S  x 
20.8  cm.,  the  text  being  9.8  x  16  cm.  35  ff.,  34  11.  Paris  1571.  The 
Proeme  is  dated  "  De  la  grolTe  tour  de  Bourges  le  douziefme  d'o-// 
ctobre.  1571."    An  arithmetic  for  instruction  in  military  circles. 

Page  347.  A  1590  Antwerp  edition  of  Menher  has  been  acquired: 

*  Livre // d'Arithme-//tique  contenant  plufieurs  belles  que-//ftions  & 
demandes,  bien  propres  &  // vtiles  a  tous  Marchans,  //  Par  M.  Valentin 
Mennher  //de  Kempten.  //  Reueu  par  M.  Melchior  van  Eiftaer.//  . . .' 

Page  375.  The  1584  Cologne  edition  of  Clavius  has  been  acquired. 

Page  383.  '  Arithmetica  //  oder  //  Rechenbuch  //  .  . .  durch  //  An- 
thonium  Schultzen/. .  .  Zur  Liegnitz  .  .  .  1600.'  FoL,  13.6  x  18.5  cm., 
the  text  being    10.2  x  13.5  cm.    4  +  259!^.,   besides  an   appendix  on 

*  Buchhalten,'  30  11.  The  first  edition  apjieared  in  1583.  A  book  of  no 
particular  merit. 

Page  389.  In  1585  Fr.  Barocio  published  a  Cosmographia,  at  Venice. 
It  contains  5  pp.  on  arithmetic. 

Page  391.  The  work  of  Aurelio  Marinati  has  been  acquired:  'La 
prima  parte  //  della  //  somma  di  tvtte  //  le  scienze  //  nella  qvale  si  tratta 
delle//sette  arti  liberali// ...  Roma. ..  1587.'  4°,  16  x  21.5  cm.,  the 
text  being  11  x  17.7  cm.  8  +  156  pp.  The  chapter  '  Dell'  aritmetica ' 
begins  on  p.  99  and  ends  on  p.  114. 

Page  408.  Vila's  arithmetic  has  been  acquired:  '  Reglas // brevs 
de  Arith-//metica  . .  .  per  Bernat  //  Vila, . . .  Barcelona  .  .  .  Any.  M.D.- 
LXXXXVI.'   8°,  9.8  x  14.3  cm.,  the  text  being  7.5  x  12  cm.  8+i36fT. 

Page  418.  The  161 1  edition  of  Mariana's  work  has  been  acquired. 
'  loannis  //  Marianae  //  Hifpani  //,  e  socie.  lesv,  //  De  Ponderibvs  //  et 
Mensvris. //Typis  Wechelianis. // Anno  M.DC.XI.' 

Page  425.  A  manuscript  of  Zuchetta's  work,  copied  in  1692,  has 
been  acquired. 

Page  440.  There  has  been  acquired  one  folio  of  manuscript  on 
parchment,  in  a  fourteenth-century  hand,  containing  part  of  the  Ety- 
mologies of  Isidorus.    31  x  45.7  cm.    See  also  p.  8. 

Page  456.  There  has  been  acquired  an  anonymous  manuscript  on 
astrology,  with  some  directions  for  arithmetical  computations.  It  bears  no 
date  but  was  written  c.  1450.   5  fi.,  on  paper,  in  Latin,  in  a  German  hand. 

Page  480.  A  manuscript  of  Bede,  '  De  Scientia  computandi,'  written 
in   Latin,  c.   1520,  has  been  acquired.    Fol.,   22.7  x33cm.,  the  text 


RARA   ARITHMETICA 

being  13  x  22.5  cm.    In  the  same  volume  and  written  by  the  same  hand 
is  '  Franconis  De  Quadratura  Circuli  lib.  V.' 

Page  487.  A  manuscript  of  Boethius  and  Gerbert,  on  geometry, 
written  in  Latin,  c.  1550,  has  been  acquired.  The  manuscript  of  Ger- 
bert begins  •  Incipit  liber  geometrie  artis  seditvs  a  Dno // Gerberto 
Papa  et  Philosopho.  Qvi  et  Silvester  //  secvndvs  est  nominatvs.'  Each 
manuscript  contains  some  interesting  number  work.  Fol.  on  paper, 
22  X  33.5  cm.,  the  text  being  15.5  X  23.5  cm. 

As  stated  in  the  preface,  there  will  naturally  be  found  from  time  to 
time  numerous  additions  to  the  bibliography  contained  in  this  work. 
The  following  have  recently  come  to  the  attention  of  the  author : 

Page  70.  Portius.    There  was  also  an  edition,  Rome,  1524,  4°. 

Page  81.  Boethius,  Clichtoveus,  and  Stapulensis.  There  was  also  an 
edition,  Paris,  15 14,  fol. 

Page  97.  There  is  an  anonymous  '  Algorithmus  linealis,'  1513  at 
Gottingen,  —  probably  Licht. 

Page  106.  Kobel.  There  was  an  edition  of  the  '  Vysierbuch,'  Oppen- 
heim,  15 19. 

Page  123.  Grammateus.   There  was  also  an  edition,  Frankfort,  1554. 

Page  139.  Riese.  There  was  also  an  edition,  1559,  and  a  Frankfort 
edition,  1563,  of  No.  2. 

Page  180.  Albert.    There  was  also  an  edition,  Magdeburg,  1588. 

Page  195.  Noviomagus.   There  was  also  an  edition,  Cologne,  1539. 

Page  214.  Recorde.   There  was  also  an  edition,  London,  1658. 

Page  257.  Giilfiferich.    There  was  also  an  edition,  Frankfort,  1561. 

Page  286.  There  was  published  a  small  Tariffa  by  Marcello  in  1566. 

Page  359.  There  may  have  been  a  Rechenbuch  by  Junge  published 

in  1577- 
Page  396.  Lindebergius  published  at  Rostock,  in  1591,  a  work  '  De 
prsecipuorum  .  .  .,'  containing  a  little  arithmetic. 

Page  429.  There  is  also  a  work  by  Lachar,  '  Algorithmus  mercatorum,' 
s.  1.  a. 

The  statement  in  the  preface,  page  ix,  relating  to  the  De  Morgan 
library  should  be  modified.  While  some  of  the  books  were  sold  (for 
example,  see  page  122),  most  of  them  were  purchased  by  Lord  Over- 
stone  and  presented  to  the  University  of  London,  where  they  may  now 
be  seen. 


INDEXES 


1472-80 

8,  9,  .c 

.478 

3 

.48. 

.0 

i4Sz 

1  1,  .2 

■483 

13.  15 

14S4 

15.  '8 

1485-7 

23 

1488 

25.36 

1489 

36,  39 

.490 

4I'  44 

1 49 1 

47,  49 

.492 

50-  54 

1493 

54 

1494 

54-  56 

1495 

58,  60 

.496 

62,  63 

1497 

64 

.498 

64 

1499 

66,67 

1500 

70,  71 

.501 

71,76 

.502 

76 

1503 

77,83 

1504 

83 

1505 

84,  86 

.506 

86 

1507 

86,  87 

1508 

87 

1509 

87,  91 

1510 

89,91 

'5" 

91 

1512 

9',  93 

■5'3 

94.  97 

15.4 

98,  .06 

INDEX  OF  DATES 


DATES   OF    PRINTED    BOOKS 

15.5  .14,  122 

.5.6  .22 

.517  122,  123 

1518  123,  126 

1 519  .26,  127 

1520  .27,  128 

152.  .31,  132 

1522  132,  .40 

1523  140 

1524  .40 

1525  .40 

1526  143,  152 

1527  153.156 
152S  157,  159 

1529  159 

1530  159,  164 

153.  165,167 

1532  .68 

1533  lyt-  >73 

1534  .74,  180 

1535  180,181 

1536  181,  182 
.537  183,  186 

1538  186,  188 

1539  19'.  '95 

1540  197,2.. 
.54.  2.1 

.542  2.2,  216 

.543  221,223 

1544  223,229 

1545  231,  238 
.546  340,243 

1547  244 

1548  244 

497 


■549 

245.  249 

'550 

249.  252 

'55' 

252.  253 

'552 

254,  257 

'553 

257,  260 

1554 

260,  263 

'555 

263,  269 

'556 

27.,  2S6 

'557 

286,  290 

'558 

290,  292 

'559 

292 

.560 

295,  298 

,56. 

298,  306 

.562 

308,3.. 

'563 

3"-3'4 

1564 

3>5 

'565 

3'6,  3'9 

is66 

320,  322 

.567 

325 

.568 

325.  328 

.569 

330,  338 

'570 

338 

'57' 

338 

1572 

340,  343 

'573 

343.  346 

1574 

346 

'575 

348,  352 

1576 

353 

1577 

353.  359 

'578 

359.  36' 

•579 

36.,  364 

.580 

364,  367 

.58. 

36S 

'582 

36S,  375 

498  RARA   ARITHMETICA 


1583 

375'  380 

15S9 

392.  393 

1595 

407 

1584 

380,  383 

1590 

393 

1596 

407,  40S 

1585 

385'  389 

1591 

394-  396 

1597 

408 

1586 

389 

'592 

396,  400 

1598 

409,  415 

1587 

389.  391 

1593 

404 

1599 

415 

I58S 

391 

1594 

404,  407 

1600 

425,  427 

DATES   OF   MANUSCRIPTS 


1260 

433 

1447 

45' 

1 501 

480 

1294 

434 

1450 

45- 

1522 

480 

1300 

435 

1456 

454.  458 

i525 

481 

1339 

435 

1460 

459 

1533 

482 

1350 

440 

1462 

465 

1535 

482 

1375 

442 

1469 

466 

1545 

484 

1384 

443 

1473 

466 

1550 

487 

1393 

443 

1475 

468 

1560 

488 

1400-35 

439 

1476 

470 

1565 

488 

1422 

443 

1477 

473 

1568 

490 

1424 

446 

1478 

473 

1575 

490 

1430 

447 

1488 

474 

1579 

492 

1435 

449 

1490 

475 

1598 

493 

I44I 

449 

1492 

477 

1599 

493 

1442 

458 

1500 

477 

1600 

493 

INDEX  OF  NAMES,  PLACES,  AND 
SUBJECTS 


Abacus,  7.  70.     See  Counters 

Adriaen  van  der  Gucht,  338 

Agricola,  i  7 1 

Alamagni,  404 

Albert,  J.,  17S 

Albert  of  Saxony,  9,  3,  465 

Albertus  Magnus,  442 

Alcala,  60,  167,  231,  310,322 

Alciatus,  164 

Alexandrinus,  389 

Algebra,  56,  125,  132,  226,  233, 

260,  2S6,  400,  403,  454 
Algebra  (facsimile),  455 
Algebraic  symbols  (facsimile), 


Algorism,  5,  7,  10,  13,15,  28,  33^ 
450,  454 

Al-Khowarazmi,  7,  454 

Alkmaar,  325 

Alligation,  iS 

Almadiano,  23 

Alonso  Delatore,  41 

America,  286 

Ammonius,  367,  429 

Amsterdam,  325,  424 

Andrea  Lanfreducci,  475 

Andres,  122 

Andres  Garcia  de  Lovas,  253 

Angelus  Mutinens,  140 

Anianus,  31,7 

Annaberg,  139 

Anonymous,  3,  23,  64,  67,  71 
106,  120,  126,  127,  128,  130, 
178,  186,  195,  211,  212,  223, 
252,  286,311,338,340,359, 
408,  429,  443,  447,  449.  45O' 
456,  458,  459,  463,  464,  466, 
473.475.  477.478,480,482, 
493.  494 


Antonius  de  Barres,  238 

Antwerp,  32,  183,  200,  249,  250,  278,  281, 
3'5.  345.  365.  372,  375.  38"6,  427 

Apianus,  155,  62 

Aquila,  396 

Arabic  numerals,  1 1 1,  447 

Arabic  numerals  (facsimiles),  447,  I'l. 
VIII,  et  passim 

Archimedes,  226 

Aristotle,  15 
252,254,       Arithmomachia,  1  2.    .SVf  Rithmomachia 

Arnheim,  iSS 

Astrolabe,  72 
403  Astrological  numerals,  375 

74,  106,       Astronomical  fractions.    Si-e  Sexagesi- 
mals 

Augsburg,  8,  10,  25,  27,  37,  41,  100,  102, 
119,  130,  152,  160,  168,  186,  190,  238, 
263,  322,  3?>^ 

Augustine,  83 

Aurel,  254 

Austria.     See  Cracow,  Vienna,  etc. 

Autorff,  250 

Aventinus,  136,  229 

Avignon,  130 


Bachet,  34S 

Basda,  131,  140,  159 

Baeza,  269 

Baker,  327 

I,  83,  86, 

Bamberg,  12,  15 

153.  173. 

Barcelona,  93,  310,  316,  375,  40S 

243,  244. 

Barchi,  263 

368,  389, 

Barlaamo,  315,  343 

452.  454, 

Barocius  (Barozzi),  295,  340 

468,  470, 

Barres,  238 

486,  488, 

Barter,  18,  399 

Bartoli,  315 

499 


500 


KARA   ARITHMETICA 


384 


Barziza,  140 

Basel,  8,  II,  27,  43,  60,  66,  70,  82,  86, 
131,  164,  165,  166,  167,  168,  171,  180, 
182,  186,  192,  211,  226,  246,  265,  266, 
269,  286,  315,  33O'  335.  338.  348,  356, 
361,  372,  375'  389'  407.  \-l 
Beausard,  346 
Bede.    See  Baeda 
Beldamandi,  13 

Belgium.     Sec  Antwerp,  Bruges,  etc. 
Belli,  343 
Benedetti,  364 
Benedetto  da  Firenze,  464 
Benese,  182 
Benicansa,  402 
Berenguer,  359 
Bergamo,  37  5,  380,  382 
Bernard,  452 
Bertholio,  263 
Bilstenius,  407 
Blasius,  95 
Blundevile,  407 
Bock,  229,  249 

Boethius,  25,  62,  80,  4,  13,  82,  434 
Bogardus,  229 
Boissiere,  260,  271 
Bologna,  61,  244,  253,  295,  356,  359 
Bonini,  122 
Bonocchio,  347 
Borghetti,  407 
Borghi,  16 
Borriglione,  86 
Boschensteyn,  A.,  182 
Boschensteyn,  J.,  100 
Boscherus,  393 
Bouvelles  (Bovillus),  89 
Bradwardin,  61,  5,  86,  117,  451 
Brandt,  169,  408 
Brasser,  393 
Braunschweig,  391 
Brescia,  325,  347,  368,  425 
Breslau,  139,  322,  347,  404,  412 
Bruc3eus,  350 
Bruges,  338 
Bruno,  396 
Brunus,  396 
Buckley,  252 
Budaeus.  99,  35, 
Budapesth,  67 
Bude.     See  Budaeus 


164 


Budel,  396 
Bungus,  380 
Buscher,  393 
Buschius,  86,  106 
Buteo,  292 

Caesar  of  Padua,  368,  402 

Calandri,  47 

Calculi.     See  Counters 

Calendar.     See  Computus 

Calvino,  76 

Camerarius,  262,  186,  211 

Camilla,  315 

Campanus,  433 

Canacci,  459 

Cantone,  416 

Capella,  66 

Cardanus,  193,  338 

Cardanus  (portrait),  194 

Casanova,  292 

Cassiodorus,  2ii 

Cataldi,  356 

Cataneo,  242 

Catechism  method,  197,  210,  290 

Catechism  method  (facsimile),  199 

Cathalan,  268 

Cattaldi,  356 

Champenois,  359 

Ch  ampler,  186 

Chauvet,  359 

Chiarini,  10,  56 

China,  429 

Chuquet,  128 

Cipro,  10 

Ciruelo,  58,  61 

Clatovenus,  292 

Clavius,  375 

Clement,  375 

Clichtoveus,  30,  80,  82,  94 

Cognet,  346,  365 

Coignet.     .5"^^  Cognet  ■ 

Cologne,  23,  27,  46,74.  76,  85,  153,  154. 
159,  167,  169,  180,  195,  200,  212,  231 
281,367,  375,  396,402,429 

Computus,  7,  31,  41,  73,  126,  443,  449 
463,  473 

Computus  (facsimiles),  444,  445'  472 

Constantinople,  180 

Copenhagen,  353 

Corella,  244 


INDEX 


501 


Cortes,  407 

Coss,  125,226,233.260,286.  .S'-ff  Algebra 
Counters,  7,  36,  155,  271,  412,  4cS2 
Counters  (facsimiles),  69,  103,  156,  166, 

201,  214,  215 
Covarrubias,  396 
Cracow,  32,  97,  123,  190,  260,  303 
Cuento,  60,  249 
Culman,  182 
Cuno,  269 
Curtius,  396 

Cusa,  J.,  106.    See  also  Nicolaus  Cusa 
Czemy,  353 

Dagomari,  435 

Debreczin,  359 

Decimal  point  (facsimiles),  52,  388 

Decimals,  50,  329,  330,  386,  388 

Dee,  214 

De  la  Roche,  128 

Delfino,  275 

Denmark.    See  Copenhagen 

De  Suberville,  409 

Deventer,  64,  67,  76,  188,  195,  262 

Dialing,  4S0 

Diego  el  Castillo,  46 

Digges,  340 

Diophantus,  348 

Division,  18,  49,  449 

Division  (facsimiles)  — 
A  danda,  47,  462,  489 
Galley,  6,  100,  483,  486,  487 
Of  fractions,  485 

Division  of  fractions,  233,  262 

Dollar  sign,  470 

Dollar  sign  (facsimiles),  471,  491 

Domenicho  de  Valsugana,  492 

Dordrecht  (Dort),  392 

Dortmund,  213 

Duni,  269 

Duplation,  75 

Diisseldorf,  300 

Dycke,  427 

Eclipse  (facsimile),  453 

Eisenmann,  91 

Eisenmenger,  3S9 

Eisleben,  303 

Elias  Misrachi,  180 

England,  134,  216,327,450.    See  London 


Eciuality  sign  (facsimile),  288 

Equation  of  payments,  439,  446 

Erfurt,  124,  139,  33S 

Eschenburg,  340 

Espinosa,  238 

Euclid,    II,  236,   4,    56,  433,  435,  440, 

442,  463,  481,  486 
Euclid  (facsimiles),  436,  PI.  IV,  VI 
Everardus,  396 
Exchange,  38 
Exchange  (facsimile),  38 
Eysenhut,  190 

P"aber  Stapulensis,  62,  80,  27,  30,  82 

False  position,  18 

Feliciano,  146 

Fermat,  348 

Fernandez  de  Santaella,  140,  269 

Fernelius,  157 

Ferrara,  143,  146,  394 

Figurate  numbers  (facsimiles),  27,  I'l.  I 

Finaeus,  160,  279,  82,  97,  106 

Finger  symbols,  34,  57,  95,  115,  121, 
131.  137,  257 

Finger  symbols  (facsimiles),  57,  138 

Fischer,  247 

Flicker,  367 

Florence,  10,  15,  16,  47,  63,  70,  98,  99, 
122,  132,  140,  346 

FcEniseca,  119 

Fonduli,  295 

Fonduli  (portrait),  296 

Forcadel,  284,  316 

Fractions,  106.  See  Decimals,  Sexa- 
gesimals, Division 

France,  284.    See  Paris,  Lyons,  Avignon 

Francesco  dal  Sole,  143 

Franciscus,  15 

Franci.scus  de  Oretio,  396 

Frankfort  a.  ^L,  32,  53,  102,  106,  123, 
139.  154.  '55.  J(^9.  iSo.  200,  244,  253, 
269,  290,  295,  298,  319,  330,  331,  340, 
359.  368,  3^3'  393'  396,  408,  429 

Frankfort  a.  d.  Oder,  139,  247,  289 

Frascada,  490 

Freiburg,  82,  181,  1S2,  192,  391,  410 

Freigius,  372 

Frey,  Jacob,  338 

Frey,  Johann,  221 
Freyle,  286 


502 

Fulconis,  389 
Furst,  93 


KARA   ARITHMETICA 


Galasso,  404     . 

Gauging,  114.  M^,  221,  468 

Gauging  (facsimiles)  113,  222 

Gehrl,  359 

Gematria,  404 

Gemma  Frisius,  200 

Gempelius,  391 

Genoa,  416,  425 

Gentil,  263 

George  of  Hungary,  67 

Germany.      Sec    Augsburg,     Bamberg, 

Heidelberg,  etc. 
Gersbach,  353 
Getons.     See  Counters 
Ghaligai,  132 
Ghebelino,  325 
Giovanni  da  Firenze,  443 
Girard,  386 

Gifjka  Gorla  z  Gorlssteyna,  353 
Giuntini,  492 
Glareanus,  191 
Gleitsmann,  427 
Glysonius,  408 
Gmunden,  117,  449 
Goldammer,  269 
Gosselin,  279 

Gouda,  386,  394,  424 

Grammateus,  123 

Granada,  93 

Gray,  353 

Gualda,  167 

Gucht,  338 

GUlfferich,  269,  292 

Gullucci,  82 

Gutierrez  de  Gualda,  167 

Guyion,  130 

Gyraldus,  254 

Haarlem,  421 
Hagenau,  37,  76,  164 
Halle,  303 
Hamburg,  393 
Hanaellius,  228 
Hangest,  87 
Hartwell,  220 
Heere,  421 
Hegelin,  229 


Heidelberg,  168,  266,  356,  359 

Heller,  67 

Helm,  141,  142,  319 

Helmreich,  303 

Helmstadt,  393 

Henry-metre,  409 

Herbestus,  303 

Herodianus,  60 

Hobel,  314 

Hoflein,  389 

Holland.  .5"^^ Amsterdam,  Deventer,  etc. 

Honoratus,  487 

Hood,  361 

Horem,  86,  117 

Hornmannus,  396 

Horoscope,  478,  492 

Hostus,  372 

Huguetan,  188,  131 

Hungary.    See  Budapesth,  Debreczin 

Huswirt,  74 

Hiitzler,  263 

Hylles,  396 

lamblichus,  188,  186 

Ingolstadt,  155    " 

Instruments  (facsimiles),  124,  283,  297, 

367 
Isidorus,  8,  3,  10 
Italy.    See  Florence,  Venice,  etc. 

Jacob,  295 

Jacob's  staff,  no 

Jeronimo  de  Valencia,  269 

Joannes  de  Gmunden,  117,  449 

Joannes  de  Muris,  117 

Johann  von  Gmunden,  117,  449 

Jordanus,  62,  5,  27,  82 

Kandleon,  368 

Kaudler,  368,  396 

Klos,  190 

Klumpius,  493 

Kobel,  100 

Kolross,  164 

Konigsberg,  226,  258,  260,  314 

Kopfer,  346 

Krafft,  402 

Lagasa,  315 
Landshut,  83,  97 


INDEX 


503 


Lanfreducci,  475 

Lange,  353 

Lanzut.    See  Landshiit 

Lapazzaia,  322 

Lapazzaia  (portrait),  323 

Laudensis,  396 

Lautenschlager,  410 

Lax,  121 

Lazesio,  146 

Leipzig,  27,  36,  37,  44,  53,  70,  83,  86,  91, 
120,  138,  139,  155,  168,  171,  200,  223, 
233,  247,  249,  250,  263,  290,  303,  31 1, 
331.  340,  353,  361,  365,  379,  412,  421 

Lemgo,  330 

Leonardus  Maynardus,  474 

Leunbach.  86 

Leyden,  66.  165,  166,  210,  254.386.  See 
Lugduiii 

Licht,  70 

Liegnitz,  383 

Lillius,  63 

Liverius,  13 

London,  10,  132,  134,  182,  195,  213,  214, 

244,  252,  253,  286,  327,  330,  338,  340, 
353.  368,  391.  393.  396,  400,  407,  408 

Lonicerus,  253 

Lopez  de  Corella,  244 

Loritus,  191 

Lortze,  91 

Lossius,  289 

Letter,  70 

Lottini,  2S6 

Louvain,  238,  346 

Lovas,  253 

Liibeck,  263,  393 

Luca  da  Firenze,  468 

Luca  de  Borgo,  54,  87 

Lucar,  393 

Ludovico  Alt,  4S4 

Lugduni  (Lyons  or  Leyden),   167,  200, 

286 
Lullius,  415,  457 
Luyck,  408 
Lyons,  32,  93,   128,  130,  165,  171,    192, 

245,  253,  269,  286,  290,  292,  314,  320, 
385,  389.     See  Lugduni 

Madrid,  310,  393,  404 

Maffei,  86 

Magdeburg,  139,  180,  290,  389 


Maginus,  399 

Mainz,  23,  118 

Mameranus,  396 

Manenti,  174 

Mantua,  66,  263 

Manzoni,  257 

Mariana,  418 

Mariani,  180 

Marinati,  391 

Martin,  231 

Masterson,  400 

Matiieniatics,  value  of,  72,  127,  231,  300 

Maurolycus,  348,  350 

Maynardus,  474 

Measures,  70,  99,  164,  171,  209,  266,  319 

Mediate,  75 

Medina,  229 

Medlerus,  223 

Melanchthon,  210,  53,  164,  226,  236 

Melero,  181 

Mellema,  375 

Mellis,  218,  214,  391 

Menher,  249,  281,  346 

Menochius,  396 

Mensuration,  478 

Mercado,  335 

Mercatello,  480 

Meres,  408 

Messina,  93,  429 

Mewrer,  396 

Mexico,  286 

Meyer,  322 

Micyllus,  265 

Middelburg,  418 

Milan,  76,  114,  115,  193,  216 

Military  arithmetic,  343,  359,  385 

Million,  17 

Million  (facsimile),  19 

Mirandula,  127 

Misrachi,  180 

Modena,  66 

Mohammed  ibn  Musa,  7,  454 

Molinaeus,  396 

Monhemius,  300 

Monte  Regal  Piedmontois,  3S5 

Monzo,  292 

Mordente,  415 

Morsianus,  159,  1S2 

Moya,  308 

Mugling,  450 


504 


RARA   ARITHMETICA 


Multiplication,  56,  115,  447 

Multiplication  (facsimiles)  — 
Complementary,  76 
t"anciful,  116 
Gelosia,  5,  448,  461 
Per  quadrato,  5,  448,  461 
Per  scachiero,  5,  6,  448,  483 
Table,  26,  37,  446,  460,  464,  479 

Multiplication  table,  37,  446 

Miinster,  85 

Miinster,  Seb.,  180 

Munyos,  320 

Muris,  1 17 

Mutinens,  140 

Mystery  of  numbers,  95,  91,  199,  383 

Nabod,  281 

Naples,  322,  324,  402,  416 

Neander,  266 

Nessen,  322 

Nevvdorffer,  415,  477 

Nicolaus  Cusa,  42 

Nicolo  de  Orbelli,  23,  473 

Nicomachus,  186,  4,  27 

Nonius,  315 

Norico,  44 

Norry,  347 

Noviomagus,  195,  159 

Numerals.      See    Arabic,    Astrological, 

Roman 
Numeration,  260 
Nuiiez,  315 
Niirnberg,  53,  62,  86,  123,  137,  139,  151, 

152,  154,  156,  160,  178,  1 82,  193,  221, 

223,  226,  231,  249,  271,  292,  300,  314, 

338,  346,  391,  412,415,421 

Obernheym,  473 
Obers,  238 
Ofenlach,  389 
Oliva,  126 

Oppenheim,  102,  106 
Oresme.    See  Horem 
Ortega,  91 
Otto,  361 

Paciuolo,  54,  87 
Padovanius,  389 
Padua,  9,  ID,  13,  150 
Paetus,  343 


Pagani,  394 

Pagnini,  308 

Paolini,  393 

Paolo  deir  Abaco,  435 

Paris,  9,  27,  32,  43,  58,  60,  61,  62,  63,  76, 
82,  86,  87,  89,  93,  95,  97,  99,  106,  121, 
126,  131,  134,  157,  160,  167,  168,  171, 
186,  191,  192,  195,  200,  211,  223,  228, 
229.  233,  237,  245,  252,  260,  263,  268, 
269,  271,  278,  279,  284,  31 1,  314,  316, 
330,  340,  343>  347.  359.  382,  409.  446, 
45° 

Parley,  408 

Parma,  382,  384 

Partnership,  18 

Pascal  triangle,  155,  236 

Pascal  triangle  (facsimile),  156 

Pauerus,  286 

Paul.    See  Paolo,  Paulus 

Paulinus,  393 

Paulus  Alexandrinus,  3S9 

Pavia,  9,  10,  295 

Paxi,  77 

Peele,  338 

Peer,  156 

Pelacani,  140 

Peletier,  245 

Pellos,  50 

Per-cent  sign,  439,  441,  458 

Per-cent  sign  (facsimiles),  437,  440,  441, 
459.  476 

Perez  de  Oliva,  126 

Perfect  number,  91 

Petri,  325 

Petrus  de  Alliaco,  41 

Peurbach,  53,  117 

Peverone,  290 

Peverone  (portrait),  291 

Pforzheim,  36 

Piccini,  346 

Picus  Mirandula,  127 

Piedmontois,  385 

Pirkheymer,  396 

Pisa,  16 

Piscator,  247 

Plus  and  minus,  40,  126,  1S3,  254,  456 

Plus  and  minus  (facsimiles),  40,  125, 
185,  234 

Poeppingius,  391 

Poitiers,  245 


INDKX 


505 


Poland,  353.     Sff  Cracow 
Fortius,  70 
Poveiano,  375 
Prag.  292,  359 
Priscian,  319 
Problems  (facsimiles) — 

Cistern,  48 

Couriers,  1 12 

Fallen  tree,  49 

Fish,  438 

Gauging,  113,  222 

Hare  and  hound,  465 

Horseshoe  nails,  4S4 

Market  women,  1 1 1 

Partnership,  402 

Snake  and  well  (tree),  48,  439 

Thief,  438 
Proportion,  46,  89,  99,  235 
Proportion  (facsimile),  28 
Prosdocimo,  13 
Psellus,  168 
Pythagoras  (portrait),  46 

Quadrans  (facsimile),  162 
Quirini,  429 

Raets,  365 

Raggius,  98 

Rainer,  200 

Ramus,  263,  330,  335 

Raphael  Franciscus,  15 

Raymundus  Lullius,  457 

Recorde,  213,  253,  286 

Recreations,  391 

Regensburg,  136,  137,  396,  368 

Reggio,  370 

Regiomontanus,  62 

Regius,  I  Si 

Regnaudus,  396 

Regnier,  200 

Reichelstain,  169 

Reinhard,  421 

Reisch,  82 

Reymers,  379 

Rheticus,  211 

Ricci,  429 

Riese,  A.,  138,  171,  250,  7 

Riese,  A.  (portrait),  251 

Riese,  I.,  252,  365 

Ringelbergius,  165 


Ringhieri,  253 

Rithmonuichia  (Kithmimachia,  Ryth- 

momachia),  12,  63,  271,  340 
Rithmomachia  (facsimiles),  64,  273 
Rizzo,  298 
Rocha,  316 
Roche,  128 

Rodolphus  Spoletanus,  122 
Rodriguez,  408 
RoUandus,  446 
Roman  numerals,  104,  106,  249,  3S2,  383, 

Roman  numerals  (facsimiles),  104,  105, 

107,  373.  382,  3^3 
Rome,  32,  86,  91,  93,  122,  140,  236,  375, 

378,  391,  415,  429 
Ros,  457 
Rostock,  350 
Roth,  493 

Rotterdam,  250,  423,  425 
Rouen,  32 
Rozino,  451 

Rudolff,  151,  159,  126,  7,  260,  493 
Ruff  us,  31 

Rule  of  three,  18,  46 
Rules,  76 

Sacrobosco,  31,  236,  450,  453 
Salamanca,  10,  216,  238,  253,  308,  315, 

335.  408 
Salignacus,  359 
Salispurgo,  484 
Santa  Cruz,  404 
Santaella,  140,  269 
Sarafino  da  Campora,  429 
Saragossa,  99,  122,   140,  167,  iSi,  247, 

249,  269,  359 
Saravia,  229 
Savonne,  314 
Scarufh,  370 

Scheubel,  233,  246,  252,  454 
Schey,  427 
Schiedam,  423 
Schleupner,  412 
Schonerus,  J.,  178 
Schonerus,  L.,  330,  32,3 
Schreckenberger,  389 
Schreckenfuchsius  (Schreckfuchs),  iSo 
Schreiber.    See  Grammateus 
Schuere,  424 


5o6 


KARA   ARITHMETICA 


Schulze,  383 
Schweder,  314 
Segura,  322 
Sekgerwitz,  347 
Series  (facsimiles),  88,  478 
Seville,  41,  122,  93 
Sexagesimals,  ^^^^^ 
Sfortunati,  174 
Sherwood,  12 
Siderocrates,  389 
Siena,  367 
Silicius,  106 
Slate,  13 

Snellius,  330,  333 
Sole,  143 
Solingen,  20S 
Solorzano,  393 

Spain,  254.     See  Alcala,  Barcelona,  etc. 
Spanlin,  271 
Sphere,  453 
Spinola,  311 
Spoletanus,  122 
St.  Albans,  186 
St.  Bernard,  452,  466 
Stephano  da  Mercatello,  480 
Stettin,  139,  247 
Stevin,  386 

Stifel,  223,  231,  258,  493 
Stigelius,  249 
Stockmans,  392 

Strasburg,  10,32,33,42,76,82,  134,  154, 
182,  197,  21 1,  233,  315,  343'  370,  389' 

415 
Strigelius,  311 
Striibe,  391 
Suberville,  409 
Substractio,  97 
Suevus,  404 
Suiseth,  10,  86 
Supputandi,  De  Arte,  134 
Swinshead.    See  Suiseth 
Switzerland.     See  Basel 

Tables.    385,  400.     See  Tariffa,   Multi- 
plication 
Taf,  429 

Tagliente,  1 14,  141 
Tariffa,  77,  180,  404,  175,  181 
Tartaglia,  275 
Tartaglia  (portrait),  277 


Texeda,  240 

Theologoumena,  223 

Theoretical  books,  4 

Thierfelder,  391 

Toledo,  1.67,  418 

Tonstall,  132 

Torrentini,  76 

Toscolano,  54 

Toulouse,  348 

Tours,  168 

Trenchant,  320 

Trevisano,  408 

Treviso,  3 

Trigonometry,  474 

Tubingen,  74 

Turin,  50,  86,  364 

Turkey.     See  Constantinople 

Tzwivel,  84 

Uberti,  114 
Ulm,  229,  402 
Ulman,  391 
Unicornus,  298,  412 
Uranius,  208 
Urban  IV,  433 
Urstisius,  220,  361 

Valencia,  61,  122,  254,  292,  320,  407 

Valencia,  J.  de,  269 

Valerianus,  286 

Valla,  71 

Valladolid,  240,  244 

Valturius,  10 

Vandenbussche,  340 

Van  den  Dycke,  427 

Vander  Hoecke,  183 

Vander  Schuere,  424 

Vander  Schuere  (portrait),  422 

Vander  Wehn,  216 

Vejar,  249 

Venice,  8,  9,  10,  12,  13,  15,  16,  19,  20, 
21,  22,  27,  28,  32,  54,  60,  61,  62,  70, 
7i>  73'  77.  82,  87,  99,  114.  115'  13O' 
140,  141,  146,  148,  160,  168,  171,  173, 
174,  180,  181,  195,  200,  229,  242,  253, 
254,  257,  275,  278,  292,  298,  315,  340. 
343'  346,  347'  348,  35O'  364.  382'  389. 
393,  399,  404,  407.  408,  412 

Ventallol,  298,  319 

Vergerius,  456 


INDEX 


507 


Verini,  216 

Verona,  10,  148,  389,  404 

Verse,  399,  411,  100,  262 

Verse  (facsimile),  41 1 

Vienna,  27,  32,  53,  61,  66,  106,  1 17. 

151,  152 
Vila,  408 

Vincent  de  Heauvais,  10 
Vincento,  140 
Vincenza,  66,  375 
Vincenzo  da  Bergamo,  429 
Visconti,  368 
Vittori,  61 
Vogelin,  195 
Von  Szily,  67 
Vuelpius,  231 

Wagner,  12,  15 
Walckl,  182 
Weber,  Johann,  338 
Weights.     See  Measures 
Weissenfels,  223 


123, 


Welsch  practice,  152,  416 

Wenceslaus,  418 

Werner,  300 

Widman,  36,  40,  44 

Willichius,  197 

Willsford,  216 

Wittenberg,  32.  53,  61,   139,  168,  178, 

180,  200,  223,  236,  237,  247,  269,  286, 

367,  389,  429 
Wojewodki,  260 
Wolphius,  154 

Xy lander,  356 

Vciar,  247 

Zamberto,  481 
Zero,  76 
Zuccantini,  367 
Zuchetta,  425 
Zuchetta  (portrait),  426 
Zurich,  391,  39(J 


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